On Linear spaces of of matrices bounded rank

Hang Huang and J. M. Landsberg Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA [email protected]

Abstract.

Motivated by questions in theoretical computer science and quantum information theory, we study the classical problem of determining linear spaces of matrices of bounded rank. Spaces of bounded rank three were classified in 1983, and it has been a longstanding problem to classify spaces of bounded rank four. Before our study, no non-classical example of such a space was known. We exhibit two non-classical examples of such spaces and give the full classification of basic spaces of bounded rank four. There are exactly four such up to isomorphism. We also take steps to bring together the methods of the linear algebra community and the algebraic geometry community used to study spaces of bounded rank.

2010 Mathematics Subject Classification:

68Q17; 14L30, 15A69, 15A30

Huang supported by NSF grant DMS2302375, Landsberg supported by NSF grant AF-2203618

1. Introduction

A linear subspace $E\subset\mathbb{C}^{\mathbf{b}}{\mathord{\otimes}}\mathbb{C}^{\mathbb{c}}$ is of bounded rank if for all $e\in E$ , ${\mathrm{rank}}(e)<\operatorname{min}\{\mathbf{b},{\mathbb{c}}\}$ . We say a space of bounded rank has bounded rank $r$ if $r$ is the maximal rank of an element of $E$ . Fix ${\mathbb{a}}=\operatorname{dim}E$ .

Example 1.1.

Let $E$ be of the form $\begin{pmatrix}*&*\\ *&0\end{pmatrix}$ where the blocking is $(k_{1},\mathbf{b}-k_{1})\times(k_{2},{\mathbb{c}}-k_{2})$ . Then $E$ has bounded rank (at most) $k_{1}+k_{2}$ . These are called compression spaces, or more precisely $(k_{1},k_{2})$ -compression spaces.

Example 1.2.

Let $E=\Lambda^{2}\mathbb{C}^{2p+1}\subset\mathbb{C}^{2p+1}{\mathord{\otimes}}\mathbb{C}^{2p+1}$ be the space of $2p+1\times 2p+1$ skew symmetric matrices. Then $E$ is of bounded rank $2p$ as the rank of a skew-symmetric matrix is always even.

Example 1.3.

Let $E\subset\operatorname{Hom}(E,\Lambda^{2}E)$ be given by $e\mapsto\{v\mapsto e\wedge v\}$ , which has bounded rank ${\mathbb{a}}-1$ , the kernel of the map associated to $e$ is the line through $e$ .

We will refer to these examples as the classical spaces of bounded rank.

The study of spaces of bounded rank dates back at least to Flanders in 1962 [13], who solved a conjecture on the maximal dimension of such a space posed by Marcus. Once the problem is stated, the case ${\mathbb{a}}=2$ follows immediately from the Kronecker-Weierstrass normal form for pencils of matrices.

Atkinson-Lloyd [3] introduced the notion of primitive spaces and proposed the classification of such by rank. As was known for a long time, the case $r=1$ consists of the $(0,1)$ and $(1,0)$ compression spaces. They classified the $r=2$ case, where the only primitive space is Example 1.2 with $p=1$ . In [2] Atkinson carried out the classification for $r=3$ , the only primitive examples are Example 1.3 and its projections. In particular, there are no non-classical examples of spaces of bounded rank when $r\leq 3$ . In the same paper he observed that if one allows $r$ to be large, there are “many” such spaces. This work is reviewed in §4.1.

Sylvester [26] introduced language from geometry (vector bundles, Chern classes) to study the more restrictive problem of spaces of constant rank $r$ .

Eisenbud-Harris [12] independently introduced these tools, where for the general question one must deal with sheaves rather than vector bundles. They proposed a refinement of the classification problem to the study of basic spaces (which in particular, disallows projections of primitive spaces) and stated that they were unaware of a non-classical example of a basic space of bounded rank four. They also refined the observation that there are many such spaces for $r$ large by observing that such spaces arise as matrices appearing in linear parts of the minimal free resolutions of sufficiently general projective curves. In particular, there are nontrivial moduli of basic spaces for $r$ large. Their work is reviewed in §4.2.

To our knowledge, the first example of a non-classical space of bounded rank is due to Westwick in 1996 [28], which is even of constant rank (namely constant rank $8$ ). Since then numerous explicit examples have been found. See [19] and [23] for two recent contributions in the special case of constant rank.

Acknowledgements

We thank Austin Conner, Harm Derksen, David Eisenbud, Mark Green, Joe Harris, Laurent Manivel, Mihnea Popa, Jerzy Weyman, and Derek Wu for useful conversations. Landsberg especially thanks Harris for inviting him to Harvard where this project began to take shape.

2. Results

Basic spaces are reviewed in §4.2. All spaces of bounded rank may be deduced from the basic spaces.

Theorem 2.1.

[Main Theorem] Up to isomorphism, there are exactly four basic spaces of bounded rank four:

(I)

$E\cong\Lambda^{2}\mathbb{C}^{5}\subset\mathbb{C}^{5}{\mathord{\otimes}}\mathbb{C}^{5}$ ,
(II)

$E\cong\mathbb{C}^{5}\subset\operatorname{Hom}(\mathbb{C}^{5},\Lambda^{2}\mathbb{C}^{5})$ ,

(III)

E=\begin{pmatrix}a_{1}&&&&-a_{3}&-a_{5}\\ &a_{1}&&&-a_{4}&-a_{6}\\ &&a_{1}&&a_{2}&0\\ &&&a_{1}&0&a_{2}\\ a_{2}&0&a_{3}&a_{5}&0&0\\ 0&a_{2}&a_{4}&a_{6}&0&0\end{pmatrix}\subset\mathbb{C}^{6}{\mathord{\otimes}}\mathbb{C}^{6},

(IV)

E=\begin{pmatrix}a_{1}&a_{2}&0&0&-a_{5}&-a_{6}\\ 0&a_{1}&0&0&0&-a_{5}\\ 0&0&a_{1}&a_{2}&a_{3}&a_{4}\\ 0&0&0&a_{1}&0&a_{3}\\ a_{3}&a_{4}&a_{5}&a_{6}&0&0\\ 0&a_{3}&0&a_{5}&0&0\end{pmatrix}\subset\mathbb{C}^{6}{\mathord{\otimes}}\mathbb{C}^{6}.

The proof is given in §6. It utilizes classical (Atkinson-Lloyd) and algebreo-geometric techniques.

Question 2.2.

Eisenbud-Harris [12] show that once $r$ is large, that the basic spaces of bounded rank $r$ have moduli. What is the smallest $r$ where moduli appear?

Our new examples are part of a general construction:

Proposition 2.3.

Let $E$ be an ${\mathbb{a}}$ -dimensional space of bounded rank $r$ $\mathbf{b}\times{\mathbb{c}}$ matrices and let $F\subset\operatorname{End}(\mathbb{C}^{k})$ be an ${\mathbb{a}}$ -dimensional space of commuting matrices. Take bases of $E$ and $F$ and for each matrix of the basis of $E$ replace each entry with the matrix of the corresponding basis element of $F$ multiplied by the value of the entry to obtain a $\mathbf{b}k\times{\mathbb{c}}k$ matrix. Call the new space $\widetilde{E}$ . Then $\widetilde{E}$ is a space of bounded rank at most $kr$ .

The proof of Proposition 2.3 is given in §5.1.

The space $\widetilde{E}$ of Proposition 2.3 is interesting only if the $k\times k$ matrices are not simultaneously diagonalizable, as otherwise it is isomorphic to the direct sum of $k$ copies of $E$ .

The two new examples in the Main Theorem are when $E=\Lambda^{3}\mathbb{C}^{3}\subset\mathbb{C}^{3}{\mathord{\otimes}}\mathbb{C}^{3}$ is put in Atkinson normal form (see §5.1) and the three basis elements are respectively replaced by

\begin{pmatrix}a_{1}&0\\ 0&a_{1}\end{pmatrix},\begin{pmatrix}a_{2}&0\\ 0&a_{2}\end{pmatrix},\begin{pmatrix}a_{3}&a_{5}\\ a_{4}&a_{6}\end{pmatrix},\ \ \ {\rm and}\ \ \ \begin{pmatrix}a_{1}&a_{2}\\ 0&a_{1}\end{pmatrix},\begin{pmatrix}a_{3}&a_{4}\\ 0&a_{3}\end{pmatrix},\begin{pmatrix}a_{5}&a_{6}\\ 0&a_{5}\end{pmatrix}.

Remark 2.4.

A more geometric construction that is closely related to the construction of Proposition 2.3 is presented in Proposition 3.1 below.

Remark 2.5.

Yet another method for constructing larger spaces of bounded rank from smaller ones is given in [19, §5].

We prove several technical results about invariants of spaces of bounded rank. We introduce Atkinson invariants which arise naturally in the study of Atkinson normal form for spaces of bounded rank, and show that they upper bound the first Chern classes of the sheaves introduced in [12] (Proposition 5.7). We correct a misconception in [12] about these sheaves (Proposition 5.9, Example 5.8). In the motivation described in §3.1, tensors of minimal border rank play a role. We show that such tensors cannot give rise to interesting spaces of corank one (Corollary 5.5). We exhibit new families of corank two spaces generalizing ((III)), ((IV)) in §7, as well as several variants.

Overview

In §3 we explain the correspondence between spaces of matrices and tensors, give background information on the geometry of tensors, analyze the geometry of the tensors associated to the examples in Theorem 2.1, and give several generalizations. In §4 we review the work of Atkinson-Llyod, Atkinson, and Eisenbud-Harris. In §5 we take steps to unite the linear algebra and algebraic geometry perspectives. In §6 we prove Theorem 2.1. In §7 we provide a few additional examples of spaces of bounded rank.

3. Interpretations and generalizations via the associated tensors

3.1. Background

Throughout this article, $A,B,C$ are complex vector spaces of dimensions ${\mathbb{a}},\mathbf{b},{\mathbb{c}}$ . We let $\{a_{i}\}$ , $\{b_{j}\}$ , $\{c_{k}\}$ respectively be bases of $A,B,C$ . There is a 1-1 correspondence between tensors $T\in A{\mathord{\otimes}}B{\mathord{\otimes}}C$ , up to $GL(A)\times GL(B)\times GL(C)$ equivalence where the induced map $T_{A}:A^{*}\rightarrow B{\mathord{\otimes}}C$ is injective, and ${\mathbb{a}}$ -dimensional linear subspaces of $B{\mathord{\otimes}}C$ , i.e., points of the Grassmanian $G({\mathbb{a}},B{\mathord{\otimes}}C)$ , up to $GL(B)\times GL(C)$ equivalence, given by $T\mapsto T(A^{*})$ .

In the study of tensors, the tensors that are the least understood are those that are $1$ -degenerate: those where $T(A^{*})\subset B{\mathord{\otimes}}C$ , $T(B^{*})\subset A{\mathord{\otimes}}C$ and $T(C^{*})\subset A{\mathord{\otimes}}B$ are of bounded rank.

A tensor $T\in A{\mathord{\otimes}}B{\mathord{\otimes}}C$ has (tensor) rank one if it is of the form $T=a{\mathord{\otimes}}b{\mathord{\otimes}}c$ . The rank of $T$ is the smallest $r$ such that $T$ may be written as a sum of $r$ rank one elements, and the border rank of $T$ , $\underline{\mathbf{R}}(T)$ is the smallest $r$ such that $T$ is a limit of tensors of rank $r$ . In geometric language, $\underline{\mathbf{R}}(T)$ is the smallest $r$ such that $[T]\in\sigma_{r}(Seg(\mathbb{P}A\times\mathbb{P}B\times\mathbb{P}C))\subset\mathbb{P}(A{\mathord{\otimes}}B{\mathord{\otimes}}C)$ , the $r$ -the secant variety of the Segre variety. If $T_{A}:A^{*}\rightarrow B{\mathord{\otimes}}C$ is injective, then $\underline{\mathbf{R}}(T)\geq{\mathbb{a}}$ . The tensor $T$ is called concise when all three such maps are injective. When $T$ is concise, if it has border rank $\operatorname{max}\{{\mathbb{a}},\mathbf{b},{\mathbb{c}}\}$ , one says that $T$ has minimal border rank.

3.2. Motivations for this project

In addition to being a classical problem of interest in its own right, two motivations for this project are as follows:

Strassen’s laser method

The problem to either unblock Strassen’s laser method for upper bounding the exponent of matrix multiplication [24], or prove it has exhausted its utility, began in [1]. The problem is to find new tensors to use in the method that can prove better upper bounds on the exponent than the big Coppersmith-Winograd tensor, or prove that no such exist. Minimal border rank $1$ -degenerate tensors are a class that are not yet known to have barriers to improving the method [4].

Geometric rank and cost v. value in quantum information theory

While the rank and border rank of tensors have been studied for a long time, recently attention has been paid to the notions of subrank, border subrank, and the closely related notions of slice rank [27] and geometric rank [16]. Spaces of bounded rank give rise to tensors with degenerate geometric rank. In this paper we expand upon the (at the time surprising) result of [14], where upper bounds on geometric rank imply lower bounds on border rank.

3.3. Brief discussion of tensors and their geometry

Given a tensor $T\in A{\mathord{\otimes}}B{\mathord{\otimes}}C$ , define its (extended) symmetry group

\widehat{G}_{T}=\{g\in GL(A)\times GL(B)\times GL(C)\mid g\cdot T=T\}

and the corresponding symmetry Lie algebra $\widehat{\mathfrak{g}}_{T}$ . The actual symmetry group $G_{T}$ has dimension two less as the map $GL(A)\times GL(B)\times GL(C)\rightarrow GL(A{\mathord{\otimes}}B{\mathord{\otimes}}C)$ has a two dimensional kernel.

Several important (classes of) tensors are:

•

The unit tensor: $M_{\langle 1\rangle}^{\oplus m}\in\mathbb{C}^{m}{\mathord{\otimes}}\mathbb{C}^{m}{\mathord{\otimes}}\mathbb{C}^{m}=A{\mathord{\otimes}}B{\mathord{\otimes}}C$ , where $M_{\langle 1\rangle}^{\oplus m}=\sum_{j=1}^{m}a_{j}{\mathord{\otimes}}b_{j}{\mathord{\otimes}}c_{j}$ .
•

The $W$ -state, also known as a general tangent vector to the Segre variety: $W=a_{1}{\mathord{\otimes}}b_{1}{\mathord{\otimes}}c_{2}+a_{1}{\mathord{\otimes}}b_{2}{\mathord{\otimes}}c_{1}+a_{2}{\mathord{\otimes}}b_{1}{\mathord{\otimes}}c_{1}\in\mathbb{C}^{2}{\mathord{\otimes}}\mathbb{C}^{2}{\mathord{\otimes}}\mathbb{C}^{2}$ .
•

Given an algebra ${\mathcal{A}}$ , one may form its structure tensor $T_{{\mathcal{A}}}$ obtained from the bilinear map ${\mathcal{A}}\times{\mathcal{A}}\rightarrow{\mathcal{A}}$ given by multiplication.
•

The matrix multiplication tensor $M_{\langle\mathbb{n}\rangle}\in\mathbb{C}^{n^{2}}{\mathord{\otimes}}\mathbb{C}^{n^{2}}{\mathord{\otimes}}\mathbb{C}^{n^{2}}$ is the special case of $T_{{\mathcal{A}}}$ when ${\mathcal{A}}$ is the algebra of $n\times n$ matrices.
•

Let $G\subset GL(A)\times GL(B)\times GL(C)$ be a subgroup and let $L\subset A{\mathord{\otimes}}B{\mathord{\otimes}}C$ be a trivial $G$ -submodule. Any nonzero $T\in L$ is a $G$ -invariant tensor, i.e., $G\subseteq\widehat{G}_{T}$ . Such are particularly interesting when $G$ is large. However, already when $G$ is a regular one-dimensional torus these tensors (called tight tensors in this case) can have interesting properties. Strassen conjectures that such tensors have minimal asymptotic rank. See, e.g., [21, 8, 25].
•

A special case is when $A=B=V$ , $C=\Lambda^{2}V^{*}$ , $G=SL(V)$ and $T\in\operatorname{Hom}(\Lambda^{2}V,V{\mathord{\otimes}}V)$ is the inclusion map. When $\operatorname{dim}V$ is odd, the corresponding tensor realizes two distinct spaces of bounded rank, which are cases (I), (II) of the main theorem when $\operatorname{dim}V=5$ . When $\operatorname{dim}V$ is even, the generalization of ((II)) is still of bounded rank.
•

A special case of the previous is when $\operatorname{dim}V=3$ , as then $\Lambda^{2}V^{*}\simeq V$ and one obtains $T_{skewcw,2}\in\Lambda^{3}\mathbb{C}^{3}\subset\mathbb{C}^{3}{\mathord{\otimes}}\mathbb{C}^{3}{\mathord{\otimes}}\mathbb{C}^{3}$ .

Another motivation for this project was to find new classes of tensors of interest. In what follows we show that the tensors (III), (IV) of Theorem 2.1 have many interesting properties and generalizations.

Given $T\in A{\mathord{\otimes}}B{\mathord{\otimes}}C$ and $T^{\prime}\in A^{\prime}{\mathord{\otimes}}B^{\prime}{\mathord{\otimes}}C^{\prime}$ , their Kronecker product is just their tensor product considered as a three-way tensor: $T\boxtimes T^{\prime}\in(A{\mathord{\otimes}}A^{\prime}){\mathord{\otimes}}(B{\mathord{\otimes}}B^{\prime}){\mathord{\otimes}}(C{\mathord{\otimes}}C^{\prime})$ .

3.4. A geometric interpretation of the tensor associated to Case (IV) and a tensor variant of Proposition 2.3

Case (IV) is a special case of the following construction:

Proposition 3.1.

Let $T\in A{\mathord{\otimes}}B{\mathord{\otimes}}C$ be such that $T(A^{*})$ has bounded rank $r$ and let $T^{\prime}\in A^{\prime}{\mathord{\otimes}}B^{\prime}{\mathord{\otimes}}C^{\prime}$ with ${\mathbb{a}}^{\prime}=\mathbf{b}^{\prime}={\mathbb{c}}^{\prime}=m$ be of minimal border rank. Then $T\boxtimes T^{\prime}(A^{*}{\mathord{\otimes}}{A^{\prime}}^{*})$ has bounded rank at most $rm$ .

Proof.

The proposition clearly holds when $T^{\prime}=M_{\langle 1\rangle}^{\oplus m}$ has rank $m$ , as then one just obtains the sum of $m$ copies of $T$ . Being of bounded rank is a Zariski closed condition and all concise minimal border rank $m$ tensors are degenerations of $M_{\langle 1\rangle}^{\oplus m}$ , i.e., in $\overline{GL_{m}\times GL_{m}\times GL_{m}\cdot M_{\langle 1\rangle}^{\oplus m}}$ . ∎

Case (IV) is the special case $T=T_{skewcw,2}$ and $T^{\prime}=W$ .

Neither construction strictly contains the other: Case (III) does not arise from the construction of Proposition 3.1 and a minimal border rank $1$ -degenerate tensor does not in general correspond to a space of commuting matrices.

3.5. A geometric interpretation of Case (III) and generalizations

The matrix multiplication tensor may be defined as follows: Let $U,V,W$ be vector spaces and let $A=U^{*}{\mathord{\otimes}}V$ , $B=V^{*}{\mathord{\otimes}}W$ , $C=W^{*}{\mathord{\otimes}}U$ . Then the (possibly rectangular) matrix multiplication tensor is the (up to scale) unique $GL(U)\times GL(V)\times GL(W)$ -tensor in $A{\mathord{\otimes}}B{\mathord{\otimes}}C$ , namely $M_{\langle U,V,W\rangle}:=\operatorname{Id}_{U}{\mathord{\otimes}}\operatorname{Id}_{V}{\mathord{\otimes}}\operatorname{Id}_{W}$ . In bases we may write $M_{\langle 2\rangle}$ as

M_{\langle 2\rangle}(A^{*})=\begin{pmatrix}x^{1}_{1}&x^{1}_{2}&&\\ x^{2}_{1}&x^{2}_{2}&&\\ &&x^{1}_{1}&x^{1}_{2}\\ &&x^{2}_{1}&x^{2}_{2}\end{pmatrix}.

Consider the following construction which may be considered as an augmentation of the matrix multiplication tensor (see §7 for an even further generalization): let $U,V,W$ be even dimensional vector spaces equipped with symplectic forms $\omega_{U},\omega_{V},\omega_{W}$ . Let $A=U{\mathord{\otimes}}V\oplus W$ , $B=V{\mathord{\otimes}}W\oplus U$ , $C=W{\mathord{\otimes}}U\oplus V$ . Note that $A{\mathord{\otimes}}B{\mathord{\otimes}}C$ has a four dimensional space of $Sp(U)\times Sp(V)\times Sp(W)$ invariant tensors with basis $\omega_{U}{\mathord{\otimes}}\omega_{V}{\mathord{\otimes}}\omega_{W}$ (which, identifying $U\simeq U^{*}$ etc. using the symplectic form, is $\operatorname{Id}_{U}{\mathord{\otimes}}\operatorname{Id}_{V}{\mathord{\otimes}}\operatorname{Id}_{W}$ ), $\omega_{U}{\mathord{\otimes}}\omega_{V}$ , $\omega_{V}{\mathord{\otimes}}\omega_{W}$ , $\omega_{U}{\mathord{\otimes}}\omega_{W}$ . Here $Sp(U)=Sp(U,\omega_{U})$ is the symplectic group preserving $\omega_{U}$ . Let

(1)

T_{UVW}=\omega_{U}{\mathord{\otimes}}\omega_{V}{\mathord{\otimes}}\omega_{W}+\omega_{U}{\mathord{\otimes}}\omega_{W}+\omega_{V}{\mathord{\otimes}}\omega_{W}

so $G_{T_{UVW}}\supset Sp(U)\times Sp(V)\times Sp(W)$ . Then when $\operatorname{dim}U=\operatorname{dim}V=\operatorname{dim}W=2$ we obtain Case (III). Explicitly, Case (III) may be rewritten, setting $\{u_{1},u_{2}\}$ a basis of $U\simeq U^{*}$ etc. and $x^{i}_{j}=u_{i}{\mathord{\otimes}}v_{j}$ ,

\begin{pmatrix}x^{1}_{1}&x^{1}_{2}&&&-w_{2}&\\ x^{2}_{1}&x^{2}_{2}&&&&-w_{2}\\ &&x^{1}_{1}&x^{1}_{2}&w_{1}&\\ &&x^{2}_{1}&x^{2}_{2}&&w_{2}\\ w_{1}&&w_{2}&&&\\ &w_{1}&&w_{2}&&\end{pmatrix}.

This construction shows in particular that Case (III) has $\mathbb{Z}_{2}$ symmetry by exchanging the $B$ and $C$ factors. Note further that the space of matrices $T(B^{*})$ is not of bounded rank. By the fundamental theorem of geometric rank [16] $T(B^{*})$ must have a highly nontransverse intersection with some $\sigma_{r}(Seg(\mathbb{P}A\times\mathbb{P}C))$ . Indeed, it intersects $\sigma_{2}(Seg(\mathbb{P}A\times\mathbb{P}C))$ in a $\mathbb{P}^{1}$ .

Recall that the quaternion algebra is isomorphic to the algebra of $2\times 2$ matrices, so its structure tensor is $M_{\langle 2\rangle}$ . There is a six dimensional algebra $\mathbb{S}$ that sits between the quaternions and the octonions, called the sextonions [18]. We thank L. Manivel for observing that Case (III) is related to the sextonions.

Proposition 3.2.

The tensor $T_{UVW}$ of (1) when $\operatorname{dim}U=\operatorname{dim}V=\operatorname{dim}W=2$ is the structure tensor of the sextonions, i.e., $T_{\mathbb{S}}(A^{*})$ is Case (III).

Proof.

This will be more transparent if we instead examine $T_{\mathbb{S}}(C^{*})$ (which is isomorphic to $T_{\mathbb{S}}(B^{*})$ ). Take bases $v_{1}{\mathord{\otimes}}w_{1},v_{1}{\mathord{\otimes}}w_{2},v_{2}{\mathord{\otimes}}w_{1},v_{2}{\mathord{\otimes}}w_{2},u_{1},u_{2}$ of $B$ and $u_{1}{\mathord{\otimes}}v_{2},u_{2}{\mathord{\otimes}}v_{2},u_{1}{\mathord{\otimes}}v_{1},u_{2}{\mathord{\otimes}}v_{1},w_{1},w_{2}$ of $A$ and write $y_{ij}=u_{i}{\mathord{\otimes}}w_{j}$ . Then

(2)

T(B^{*})=\begin{pmatrix}y_{22}&-y_{21}&&&&\\ -y_{12}&y_{11}&&&&\\ &&-y_{22}&y_{21}&&\\ &&y_{12}&-y_{11}&&\\ &-v_{2}&&v_{1}&y_{22}&y_{12}\\ v_{2}&&-v_{1}&&y_{21}&y_{11}\end{pmatrix}

On the other hand the multiplication in $\mathbb{S}\simeq U^{*}{\mathord{\otimes}}U\oplus U$ is given by [18, Def. 3.11] $(X,\mu)*(X^{\prime},\mu^{\prime})=(XX^{\prime},(\operatorname{trace}(X)\operatorname{Id}-X)\mu^{\prime}+X^{\prime}\mu)$ . Writing the matrix representing the action of $(X,\mu)$ out in bases and permuting, we obtain the result. ∎

3.6. Degeneracy loci and symmetry Lie algebras

Let $T(A^{*})\subset B{\mathord{\otimes}}C$ be of bounded rank $r$ , where in this paragraph we allow that possibly $r$ is full rank. Let $\Sigma_{A}:=\{\alpha\in A^{*}\mid{\mathrm{rank}}(T(\alpha))<r\}$ denote the degeneracy locus of the space. Let $G_{\Sigma_{A}}\subset GL(A)$ denote its symmetry group and $\mathfrak{g}_{\Sigma_{A}}\subset\mathfrak{g}\mathfrak{l}(A)$ its symmetry Lie algebra. Let $\pi_{A}(\mathfrak{g}_{T})\subset\mathfrak{g}\mathfrak{l}(A)$ denote the projection of $\mathfrak{g}_{T}$ onto the first factor. Then $\pi_{A}(\mathfrak{g}_{T})\subset\mathfrak{g}_{\Sigma_{A}}$ . Thus in general one obtains that

\mathfrak{g}_{T}\subseteq\mathfrak{g}_{\Sigma_{A}}\oplus\mathfrak{g}_{\Sigma_{B}}\oplus\mathfrak{g}_{\Sigma_{C}}.

Case (II) is a space of constant rank, and has been well-studied. Case (I) drops rank over the Grassmannian $G(2,5)\subset\mathbb{P}(\Lambda^{2}\mathbb{C}^{5})$ . (Here and below we are interested in the projective geometry of the map so we work projectively - see §4.2 for more details.) The symmetry group of $G(2,5)$ is $SL_{5}$ which is indeed the symmetry group of this tensor.

Case (IV) is $\mathbb{Z}_{3}$ -invariant as it is the Kronecker product of two $\mathbb{Z}_{3}$ -invariant tensors, namely $T_{skewcw,2}\boxtimes W$ . Here $T_{skewcw,2}\in\Lambda^{3}\mathbb{C}^{3}$ and $W\in S^{2}\mathbb{C}^{2}$ . One has $\mathfrak{g}_{T_{skewcw,2}}=\mathfrak{s}\mathfrak{l}_{3}$ and

\widehat{\mathfrak{g}}_{W}=\left\{\begin{pmatrix}\lambda_{1}&\lambda_{12}\\ 0&-\mu_{1}-\nu_{1}\end{pmatrix},\ \begin{pmatrix}\mu_{1}&\mu_{12}\\ 0&-\lambda_{1}-\nu_{1}\end{pmatrix},\ \begin{pmatrix}\nu_{1}&\nu_{12}\\ 0&-\mu_{1}-\lambda_{1}\end{pmatrix}\mid\lambda_{12}+\mu_{12}+\nu_{12}=0\right\}

Here the degeneracy locus in each factor is just a linear space, e.g. $\Sigma_{A}=\{\alpha_{1}=\alpha_{3}=\alpha_{5}=0\}$ , so its Lie algebra in each factor is the parabolic stabilizing the three plane, which has dimension $27$ . This is perhaps easier to see when we write the space as

\begin{pmatrix}X&Y\\ Y&\end{pmatrix}

where $X,Y$ are spaces of skew-symmetric $3\times 3$ matrices, and the degeneracy locus is when $Y=0$ . Explicitly we have the $21$ -dimensional Lie algebra

\widehat{\mathfrak{g}}_{T_{skewcw,2}\boxtimes W}=\left\{\begin{pmatrix}a&b\\ 0&a\end{pmatrix},\ \begin{pmatrix}a&b\\ 0&a\end{pmatrix},\ \begin{pmatrix}a&b\\ 0&a\end{pmatrix}\mid a,b\in\mathfrak{sl}_{3}\right\}\oplus(\mathrm{Id}\boxtimes\widehat{\mathfrak{g}}_{W}).

The large increase over $8+5$ (where $5=\operatorname{dim}\widehat{\mathfrak{g}}_{W}$ ) is striking.

Super-additivity of dimension of symmetry Lie algebras under Kronecker product fails for generic tensors: the Kronecker product of two generic tensors will have no nontrivial symmetries. For unit tensors $M_{\langle 1\rangle}^{\oplus k}\boxtimes M_{\langle 1\rangle}^{\oplus\ell}=M_{\langle 1\rangle}^{\oplus k\ell}$ so the (extended symmetry) jump is to $2k\ell$ compared with an expected $2k+2\ell$ .

Question 3.3.

How to characterize situations where under Kronecker product, the dimension of the symmetry Lie algebra is super-additive?

The symmetry Lie algebra of Case (III) is $20$ -dimensional, and it contains $\mathfrak{s}\mathfrak{l}_{2}^{\oplus 3}$ . Explicitly, it is

\widehat{\mathfrak{g}}_{T_{\mathbb{S}}}=\mathfrak{g}\mathfrak{l}(U)\times\mathfrak{g}\mathfrak{l}(V)\times\mathfrak{g}\mathfrak{l}(W)\oplus U{\mathord{\otimes}}V{\mathord{\otimes}}W

where on $A=U{\mathord{\otimes}}V\oplus W$ , the action of $U{\mathord{\otimes}}V{\mathord{\otimes}}W$ is $u{\mathord{\otimes}}v{\mathord{\otimes}}w.(u^{\prime}{\mathord{\otimes}}v^{\prime}+w^{\prime})=\omega_{U}(u,u^{\prime})\omega_{V}(v,v^{\prime})w$ , and on $B=V{\mathord{\otimes}}W\oplus U$ , the action is $u{\mathord{\otimes}}v{\mathord{\otimes}}w.(v^{\prime}{\mathord{\otimes}}w^{\prime}+u^{\prime})=\omega_{U}(u,u^{\prime})v{\mathord{\otimes}}w$ and the action on $C$ is $u{\mathord{\otimes}}v{\mathord{\otimes}}w.(w^{\prime}{\mathord{\otimes}}u^{\prime}+v^{\prime})=\omega_{V}(v,v^{\prime})u{\mathord{\otimes}}w$ . To see $U{\mathord{\otimes}}V{\mathord{\otimes}}W\subset\widehat{\mathfrak{g}}_{T_{\mathbb{S}}}$ , write out $T_{\mathbb{S}}$ in bases

	$\displaystyle T_{\mathbb{S}}$	$\displaystyle=(u_{1}{\mathord{\otimes}}v_{1}){\mathord{\otimes}}(v_{2}{\mathord{\otimes}}w_{1}){\mathord{\otimes}}(u_{2}{\mathord{\otimes}}w_{2})-(u_{2}{\mathord{\otimes}}v_{1}){\mathord{\otimes}}(v_{2}{\mathord{\otimes}}w_{1}){\mathord{\otimes}}(u_{1}{\mathord{\otimes}}w_{2})$
		$\displaystyle-(u_{1}{\mathord{\otimes}}v_{2}){\mathord{\otimes}}(v_{1}{\mathord{\otimes}}w_{1}){\mathord{\otimes}}(u_{2}{\mathord{\otimes}}w_{2})+(u_{2}{\mathord{\otimes}}v_{2}){\mathord{\otimes}}(v_{1}{\mathord{\otimes}}w_{1}){\mathord{\otimes}}(u_{1}{\mathord{\otimes}}w_{2})$
		$\displaystyle-(u_{1}{\mathord{\otimes}}v_{1}){\mathord{\otimes}}(v_{2}{\mathord{\otimes}}w_{2}){\mathord{\otimes}}(u_{2}{\mathord{\otimes}}w_{1})+(u_{2}{\mathord{\otimes}}v_{1}){\mathord{\otimes}}(v_{2}{\mathord{\otimes}}w_{2}){\mathord{\otimes}}(u_{1}{\mathord{\otimes}}w_{1})$
		$\displaystyle+(u_{1}{\mathord{\otimes}}v_{2}){\mathord{\otimes}}(v_{1}{\mathord{\otimes}}w_{2}){\mathord{\otimes}}(u_{2}{\mathord{\otimes}}w_{1})-(u_{2}{\mathord{\otimes}}v_{2}){\mathord{\otimes}}(v_{1}{\mathord{\otimes}}w_{2}){\mathord{\otimes}}(u_{1}{\mathord{\otimes}}w_{1})$
		$\displaystyle+w_{1}{\mathord{\otimes}}u_{1}{\mathord{\otimes}}(u_{2}{\mathord{\otimes}}w_{2})-w_{2}{\mathord{\otimes}}u_{1}{\mathord{\otimes}}(u_{2}{\mathord{\otimes}}w_{1})-w_{1}{\mathord{\otimes}}u_{2}{\mathord{\otimes}}(u_{1}{\mathord{\otimes}}w_{2})+w_{2}{\mathord{\otimes}}u_{2}{\mathord{\otimes}}(u_{1}{\mathord{\otimes}}w_{1})$
		$\displaystyle+w_{1}{\mathord{\otimes}}(v_{1}{\mathord{\otimes}}w_{2}){\mathord{\otimes}}v_{2}-w_{2}{\mathord{\otimes}}(v_{1}{\mathord{\otimes}}w_{1}){\mathord{\otimes}}v_{2}-w_{1}{\mathord{\otimes}}(v_{2}{\mathord{\otimes}}w_{2}){\mathord{\otimes}}v_{1}+w_{2}{\mathord{\otimes}}(v_{2}{\mathord{\otimes}}w_{1}){\mathord{\otimes}}v_{1}$

and apply a basis vector $u_{i}{\mathord{\otimes}}v_{j}{\mathord{\otimes}}w_{k}$ to $T_{\mathbb{S}}$ . One gets a sum of six terms that cancel in pairs. Here the degeneracy locus in $\mathbb{P}T_{\mathbb{S}}(A^{*})$ is the quadric surface $\{x^{1}_{1}x^{2}_{2}-x^{1}_{2}x^{2}_{1}\}\subset\mathbb{P}^{3}\subset\mathbb{P}A^{*}$ . The parabolic preserving this is $12+6$ dimensional and by the above calculation, the nilpotent part of the Lie algebra projected onto each factor is isomorphic to $U{\mathord{\otimes}}V{\mathord{\otimes}}W$ which is contained in the parabolic.

In the general case where $U,V,W$ are $2k$ -dimensional, one obtains a $(2k)^{2}+2k$ dimensional space of bounded rank $(2k)^{2}$ . That the space is of bounded rank is transparent from the Atkinson normal form introduced below.

Question 3.4.

This construction gives rise to a series of algebras generalizing the sextonions. What properties to these algebras have? Is there interesting representation theory associated to them? (Using the sextonions, one may construct the Lie algebra $\mathfrak{e}_{7\frac{1}{2}}$ .)

Remark 3.5.

In general, the Lie algebra of derivations of an algebra ${\mathcal{A}}$ sits inside the two factor symmetry group of its structure tensor: $\widehat{\mathfrak{g}}_{T_{{\mathcal{A}}},BC}:=\{(Y,Z)\in\mathfrak{g}\mathfrak{l}(B)\times\mathfrak{g}\mathfrak{l}(C)\mid(Y,Z).T_{{\mathcal{A}}}=0\}$ . The Lie algebra of derivations of the quaterions $\mathbb{H}$ is $\mathfrak{so}(3)\cong\mathfrak{s}\mathfrak{l}_{2}$ and that of the sextonions is an $8$ -dimensional algebra containing $\mathfrak{s}\mathfrak{l}_{2}$ as its semi-simple Levi factor.

3.7. Border ranks

We thank Austin Conner for providing the decompositions in the results below:

Proposition 3.6.

The tensor $T_{skewcw,2}\boxtimes W$ corresponding to Case (IV) has border rank nine. The following is a $\mathbb{Z}_{3}$ -invariant border rank nine decomposition:

	$\displaystyle T_{skewcw,2}\boxtimes W=$
	$\displaystyle\lim_{t\rightarrow 0}\frac{1}{t^{6}}\mathbb{Z}_{3}\cdot[$	$\displaystyle(a_{1}-\frac{3}{4}t^{2}a_{3}-\frac{3}{4}t^{3}a_{4}){\mathord{\otimes}}(-\frac{2}{3}b_{1}+tb_{2}-t^{3}b_{4}){\mathord{\otimes}}(\frac{4}{9}c_{1}-\frac{2}{3}tc_{2}-\frac{2}{3}t^{2}c_{3}+\frac{2}{3}t^{3}c_{4}+\frac{1}{3}t^{5}c_{6})$
		$\displaystyle+t^{2}(a_{1}-\frac{1}{2}ta_{2}-t^{4}a_{5}){\mathord{\otimes}}(-b_{3}+tb_{4}){\mathord{\otimes}}(c_{1}+\frac{1}{2}tc_{2}-\frac{1}{2}t^{2}c_{3}+t^{4}c_{5})$
		$\displaystyle+(\frac{2}{3}a_{1}-\frac{1}{4}t^{2}a_{3}-\frac{3}{4}t^{3}a_{4}+t^{5}a_{6}){\mathord{\otimes}}(\frac{2}{3}b_{1}-tb_{2}+t^{2}b_{3}-\frac{1}{2}t^{5}b_{6}){\mathord{\otimes}}(\frac{2}{3}c_{1}+tc_{2})].$

Here the $\mathbb{Z}_{3}$ indicates cyclically permuting the factors to obtain nine terms.

Proof.

Verifying the decomposition is a routine calculation. The lower bound follows from computing the rank of the $p=1$ Koszul flattening (see, e.g., [17, Ch. 2, §2.4]). ∎

Remark 3.7.

For any two tensors $T,T^{\prime}$ one has $\underline{\mathbf{R}}(T\boxtimes T^{\prime})\leq\underline{\mathbf{R}}(T)\underline{\mathbf{R}}(T^{\prime})$ . Proposition 3.6 shows $9=\underline{\mathbf{R}}(T_{skewcw,2}\boxtimes W)<\underline{\mathbf{R}}(T_{skewcw,2})\underline{\mathbf{R}}(W)=10$ . Examples where strict submultiplicativity of border rank under Kronecker product are of interest and also potentially useful for proving upper bounds on the exponent of matrix multiplication. In particular $T_{skewcw,2}$ could potentially be used to prove the exponent is two, and strict submultiplicativity of its Kronecker square has already been observed [9]. We are currently examining which other tensors exhibit strict submultiplicativity under Kronecker product with $W$ to potentially advance the laser method.

Proposition 3.8.

The tensor $T_{\mathbb{S}}$ associated to Case (III) has border rank $10$ .

Proof.

The upper bound comes from the following decomposition, which is easily verified:

	$\displaystyle T_{\mathbb{S}}=$
	$\displaystyle\lim_{t\rightarrow 0}\frac{1}{t^{5}}[$	$\displaystyle(a_{1}-t^{2}a_{3}+t^{4}a_{6}){\mathord{\otimes}}(tb_{2}+t^{3}b_{3}+t^{4}b_{4}-tb_{5}+b_{6}){\mathord{\otimes}}(tc_{4}-c_{6})$
	$\displaystyle+$	$\displaystyle(a_{1}+ta_{2}){\mathord{\otimes}}(t^{4}b_{4}+b_{6}){\mathord{\otimes}}(-t^{3}c_{1}+\frac{1}{2}t^{2}c_{3}+c_{6})$
	$\displaystyle+$	$\displaystyle(-a_{1}+t^{2}a_{3}){\mathord{\otimes}}(t^{2}b_{1}+tb_{2}+t^{3}b_{3}+b_{6}){\mathord{\otimes}}(-t^{4}c_{2}+tc_{4}-tc_{5}-c_{6})$
	$\displaystyle+$	$\displaystyle(-a_{1}+t^{4}a_{6}){\mathord{\otimes}}(tb_{5}-b_{6}){\mathord{\otimes}}(t^{3}c_{1}-t^{4}c_{2}-\frac{1}{2}t^{2}c_{3}-c_{6})]$
	$\displaystyle+\frac{1}{t^{4}}[$	$\displaystyle(-a_{1}+ta_{2}+t^{3}a_{5}){\mathord{\otimes}}b_{5}{\mathord{\otimes}}(-\frac{1}{2}tc_{5}+c_{6})$
	$\displaystyle+$	$\displaystyle a_{2}{\mathord{\otimes}}(tb_{5}+b_{6}){\mathord{\otimes}}(t^{3}c_{1}+\frac{1}{2}tc_{5}-c_{6})$
	$\displaystyle+$	$\displaystyle(a_{1}-t^{2}a_{3}+t^{3}a_{5}){\mathord{\otimes}}(tb_{1}+b_{5}){\mathord{\otimes}}(t^{3}c_{1}-t^{2}c_{3}+tc_{4}-c_{6})$
	$\displaystyle-$	$\displaystyle(a_{1}-t^{2}a_{3}+t^{3}a_{4}){\mathord{\otimes}}(tb_{2}+b_{6}){\mathord{\otimes}}c_{5}]$
	$\displaystyle+\frac{1}{t^{3}}[$	$\displaystyle(a_{1}-t^{3}a_{5}){\mathord{\otimes}}(-tb_{1}-t^{2}b_{3}-\frac{1}{2}b_{5}){\mathord{\otimes}}(2t^{3}c_{2}-tc_{3}+c_{5})$
	$\displaystyle+$	$\displaystyle(-a_{2}+2t^{2}a_{4}){\mathord{\otimes}}(-t^{3}b_{3}+\frac{1}{2}b_{6}){\mathord{\otimes}}(-2t^{3}c_{2}+tc_{3}+c_{5})].$

To obtain the lower bound we use the border substitution method [20, 22]. Note already from Strassen’s commutation conditions applied to $T_{\mathbb{S}}(B^{*})$ , after changing bases so one basis element is the identity and using it to identify $T_{\mathbb{S}}(B^{*})$ as a space of endomorphisms, the commutator has full rank which implies $\underline{\mathbf{R}}(T_{\mathbb{S}})\geq 9$ . The border substitution method says that if for every Borel fixed hyperplane the commutator still has full rank, one obtains an improvement of one in the estimate. Here there are two Borel fixed hyperplanes, obtained respectively by setting $v_{2}=0$ and $y_{22}=0$ . In both cases the commutator is still of full rank. ∎

4. Review of previous work

4.1. Work of Atkinson-Llyod and Atkinson [3, 2]

Given $T\in A{\mathord{\otimes}}B{\mathord{\otimes}}C$ with $E=T(A^{*})\subset B{\mathord{\otimes}}C$ of bounded rank, to make connections with existing literature we write, for $\alpha\in A^{*}$ , $\phi(\alpha)=\phi_{T}(\alpha):B^{*}\rightarrow C$ for the induced map.

Consider compression spaces of bounded rank $r$ of Example 1.1: they may be described invariantly as: there exist $B^{\prime}\subset B^{*}$ of codimension $k_{1}$ , $C^{\prime}\subset C$ of dimension $k_{2}$ , with $k_{1}+k_{2}=r$ , such that $T(A^{*})(B^{\prime})\subset C^{\prime}$ . This notion is symmetric as in this case $T(A^{*})({C^{\prime}}^{\perp})\subset{B^{\prime}}^{\perp}$ .

Since compression spaces are “understood”, one would like to eliminate them from the study as well as spaces that are sums of a compression space with another space. Hence the following definition: A bounded rank $r$ space is imprimitive if there exists $H\subset B^{*}$ such that $\phi|_{H}$ is of bounded rank $r-1$ or if there exists $H\subset C^{*}$ such that $\phi^{\mathbb{t}}|_{H}$ is of bounded rank $r-1$ . In this case the bounded rank condition is inherited from the smaller space. Otherwise it is primitive.

Thus to classify spaces of bounded rank, it suffices to classify the primitive ones.

Atkinson and Llyod [3] showed that for primitive spaces of bounded rank $r$ in $B{\mathord{\otimes}}C$ , with $\mathbf{b}\leq{\mathbb{c}}$ , either $\mathbf{b}=r+1$ and ${\mathbb{c}}\leq\binom{r+1}{2}$ or there exist $r_{1},r_{2}\geq 2$ with $r_{1}+r_{2}=r$ , $\mathbf{b}\leq r_{1}+1+\binom{r_{2}+1}{2}$ and ${\mathbb{c}}\leq r_{2}+1+\binom{r_{1}+1}{2}$ .

Thus to classify primitive spaces of bounded rank $4$ , it suffices to classify them in $\mathbb{C}^{6}{\mathord{\otimes}}\mathbb{C}^{6}$ , and $\mathbb{C}^{5}{\mathord{\otimes}}\mathbb{C}^{\mathbb{c}}$ , with $5\leq{\mathbb{c}}\leq 10$ .

4.2. Work of Eisenbud and Harris [12]

Given $T\in A{\mathord{\otimes}}B{\mathord{\otimes}}C$ , with $T(A^{*})$ of bounded rank $r$ , Eisenbud and Harris observed that one gets a map between vector bundles

(3)		$\displaystyle\underline{B}^{*}$	$\displaystyle{\mathord{\;\longrightarrow\;}}\underline{C}{\mathord{\otimes}}{\mathcal{O}}_{\mathbb{P}A^{*}}(1)$
	$\displaystyle\searrow$	$\displaystyle\ \ \ \ \swarrow$
		$\displaystyle\mathbb{P}A^{*}$

The notation is that $\underline{C}:=C{\mathord{\otimes}}{\mathcal{O}}_{\mathbb{P}A^{*}}$ is the trivial vector bundle with fiber $C$ . Let ${\mathcal{E}}={\mathcal{E}}_{A}$ denote the image sheaf and ${\mathcal{F}}={\mathcal{F}}_{A}$ the image sheaf of the map with the roles of $B,C$ reversed. Since $T(A^{*})$ has bounded rank $r$ , both sheaves locally free off of a codimension at least two subset because ${\mathcal{E}},{\mathcal{F}}$ are torsion free. We slightly abuse notation, writing $\phi$ for the horizontal map (3).

They show that if $T(A^{*})$ is primitive and $c_{1}({\mathcal{E}})=1$ , then the space is obtained as a projection of the classical Example 1.3. I.e., $B\subseteq A$ and $C\subseteq\Lambda^{2}A^{*}$ .

They assert ${\mathcal{E}}^{**}\cong{\mathcal{F}}^{*}(1)$ and by symmetry ${\mathcal{F}}^{**}\cong{\mathcal{E}}^{*}(1)$ , but this is false in general, see Example 5.11. Their assertion would imply $c_{1}({\mathcal{E}}_{A})+c_{1}({\mathcal{F}}_{A})=r$ , which they do not assert but instead just assert $c_{1}({\mathcal{E}}_{A})+c_{1}({\mathcal{F}}_{A})\leq r$ , which we verify and identify the failure of equality to hold (Proposition 5.9).

Remark 4.1.

When one has a space of constant rank the assertion is correct as will be clear by Proposition 5.9 and moreover in this case ${\mathcal{E}}\cong{\mathcal{F}}^{*}(1)$ as when ${\mathcal{E}}$ is a vector bundle, ${\mathcal{E}}\cong{\mathcal{E}}^{**}$ .

In particular, to classify the first open case of $r=4$ , it suffices to understand when $c_{1}({\mathcal{E}})=c_{1}({\mathcal{F}})=2$ .

They remark that the only basic $r=4$ case they know of with $c_{1}({\mathcal{E}})=c_{1}({\mathcal{F}})=2$ is $B\cong C=\mathbb{C}^{5}$ and $A=\Lambda^{2}B\subset B{\mathord{\otimes}}B$ .

We always have $B^{*}\subseteq H^{0}({\mathcal{E}})$ and $C^{*}\subseteq H^{0}({\mathcal{F}})$ . Explicitly, the first inclusion is $\beta\mapsto([\alpha]\mapsto T(\alpha,\beta){\mathord{\otimes}}\alpha^{*}$ .

Eisenbud and Harris point out that imprimitivity is equivalent to ${\mathcal{E}}$ or ${\mathcal{F}}$ having a trivial summand. To see this, take a complement $L$ to $H\subset B^{*}$ and consider $\phi|_{L}:L\rightarrow{\mathcal{E}}$ . It is injective as $\phi$ has bounded rank $r>r-1$ , giving the desired splitting.

Note that since $\underline{H^{0}({\mathcal{E}})}$ surjects onto ${\mathcal{E}}$ , we can equally well have a trivial quotient of ${\mathcal{E}}$ , i.e., a surjection ${\mathcal{E}}\rightarrow{\mathcal{O}}$ as this induces a surjection $H^{0}({\mathcal{E}})\rightarrow{\mathcal{O}}$ which is just a linear map between vector spaces giving the splitting of ${\mathcal{E}}$ . But this in turn is equivalent to having an inclusion ${\mathcal{O}}\rightarrow{\mathcal{E}}^{*}$ , i.e., that $h^{0}({\mathcal{E}}^{*})>0$ .

In summary:

Proposition 4.2.

$E$ is primitive if and only if $h^{0}({\mathcal{E}}^{*})=h^{0}({\mathcal{F}}^{*})=0$ .

To further eliminate redundancies such as $E$ being a subspace of a primitive space or a projection of a larger primitive space, Eisenbud and Harris defined a vector space of bounded rank $T(A^{*})$ to be basic if

(1)

it is strongly indecomposable: ${\mathcal{E}}$ and ${\mathcal{F}}$ are indecomposable. This is is a strengthening of primitivity.
(2)

it is unexpandable: $\varphi$ and $\varphi^{*}(1)$ are the linear parts of kernels of maps of graded free $\mathbb{C}[A^{*}]$ modules. This says the space is not a projection of a space of bounded rank in some larger space of matrices.
(3)

it is unliftable: it is not a proper subspace of a family of the same rank in $\mathrm{Hom}(B^{*},C)$ .

Draisma [10] gives a sufficient condition for a space to be unliftable (he calls unliftability rank criticality): for $E\subset Hom(B^{*},C)$ a linear space of morphisms of generic rank $r$ , define the space of rank neutral directions

	$\displaystyle RND(E):$	$\displaystyle=\{X\in Hom(B^{*},C),\;X(Ker(Y))\subset Im(Y)\;\forall Y\in E,rank(Y)=r\}$
		$\displaystyle=\bigcap_{Y\in E,rank(Y)=r}\widehat{T}_{Y}\sigma_{r}(Seg(\mathbb{P}B\times\mathbb{P}C)).$

Here $\sigma_{r}(Seg(\mathbb{P}B\times\mathbb{P}C))$ denotes the variety of rank at most $r$ elements and $\widehat{T}_{Y}$ its affine tangent space at $Y$ .

Proposition 4.3.

[10] $RND(E)$ always contains $E$ and in case of equality, $E$ is rank critical.

5. Towards uniting the linear algebra and algebraic geometry perspectives

5.1. Atkinson normal form

Lemma 5.1.

[2, Lemma 3] Let $E\subset\mathbb{C}^{\mathbf{b}}{\mathord{\otimes}}\mathbb{C}^{\mathbb{c}}$ be of bounded rank $r$ and suppose we have chosen bases so that some matrix $X_{1}\in E$ is of the form

X_{1}=\begin{pmatrix}\operatorname{Id}_{r}&0\\ 0&0\end{pmatrix}.

Then for any $X\in E$ we have, with the same blocking,

(4)

X=\begin{pmatrix}{\mathbb{x}}&W\\ U&0\end{pmatrix}

and $U{\mathbb{x}}^{k}W=0$ for all $k\geq 0$ .

Note that it is sufficient to check $U{\mathbb{x}}^{k}W=0$ for $0\leq k\leq r-1$ because higher powers of ${\mathbb{x}}$ may be expressed as linear combinations of ${\mathbb{x}}^{0},\dotsc,{\mathbb{x}}^{r-1}$ .

Proof.

That $X$ has the bottom right block equal to zero follows by considering appropriate minors of $X+\varepsilon X_{1}$ . Consider $X_{1}+tX$ and the size $r+1$ minor

0=\operatorname{det}\begin{pmatrix}\operatorname{Id}_{r}+t{\mathbb{x}}&tW_{j}\\ tU^{k}&0\end{pmatrix}

where $W_{j}$ is the $j$ -th column of $W$ and $U^{k}$ is the $k$ -th row of $U$ . Using the formula for a block determinant and assuming $t$ is sufficiently small,

(5)

\operatorname{det}\begin{pmatrix}\operatorname{Id}_{r}+t{\mathbb{x}}&tW_{j}\\ tU^{k}&0\end{pmatrix}=t^{2}U^{k}(\operatorname{Id}_{r}+t{\mathbb{x}}){}^{-1}W_{j}=t^{2}U^{k}(\sum_{s=0}^{\infty}(-1)^{s}t^{s}{\mathbb{x}}^{s})W_{j}.

The coefficient of each power of $t$ must vanish. Since this holds for all $j,k$ , we conclude. ∎

We add the observation that the normal form is sufficient as well:

Proposition 5.2.

Let $E\subset\mathbb{C}^{\mathbf{b}}{\mathord{\otimes}}\mathbb{C}^{\mathbb{c}}$ admit a presentation in Atkinson normal form (4). Then $E$ is of bounded rank $r$ .

Proof.

The normal form assures all size $r+1$ minors containing the upper-left $r\times r$ block are zero. Say there were a size $r+1$ minor having rank $s$ in its $U$ block and $t$ in its $W$ block. We may normalize

U=\begin{pmatrix}\operatorname{Id}_{s}&0\\ 0&0\end{pmatrix},W=\begin{pmatrix}0&0\\ \operatorname{Id}_{t}&0\end{pmatrix},

where the blockings are respectively $(s,r-s)\times(s,\mathbf{b}-r-s)$ and $(t,{\mathbb{c}}-r-t)\times(r-t,t)$ . The $UW=0$ equation implies the first $s$ rows of $W$ and the last $t$ columns of $U$ are zero, so $s+t\leq r$ . Write $r+1=s+t+u$ . There must be a size $u$ minor in the ${\mathbb{x}}$ block contributing, and that minor must be in the upper right $(r-s)\times(r-t)$ block of ${\mathbb{x}}$ . But the $U{\mathbb{x}}W=0$ equation implies that the ${\mathbb{x}}$ block must be zero in the first $s$ rows of this upper right block and the last $t$ columns of this block. So the contribution must lie in a block of size $(r-s-t)\times(r-s-t)$ . But $u=r-s-t+1$ , a contradiction. ∎

Proof of Proposition 2.3.

Put $E$ in Atkinson normal form. Then $\widetilde{E}$ will also be in Atkinson normal form when the linear form times $\operatorname{Id}_{r}$ is replaced by a rank $k$ linear form. If not, just add in such a multiple of $\operatorname{Id}_{r}$ to put the space in the normal form, as this will not effect the $\widetilde{U}\widetilde{\mathbb{x}}^{j}\widetilde{W}=0$ equations. ∎

Atkinson essentially observes that the equations $U{\mathbb{x}}^{j}W=0$ may be encoded as

(6)

\begin{pmatrix}U\\ U{\mathbb{x}}\\ U{\mathbb{x}}^{2}\\ \vdots\\ U{\mathbb{x}}^{r-1}\end{pmatrix}\begin{pmatrix}W&{\mathbb{x}}W&{\mathbb{x}}^{2}W&\cdots&{\mathbb{x}}^{r-1}W\end{pmatrix}=(0),

where $(0)$ is $({\mathbb{c}}-r)\times(\mathbf{b}-r)$ . Call the first matrix on the left $At_{L}$ and call the second $At_{R}$ . Recall that if $X,Y$ are respectively $p\times r$ and $r\times q$ matrices, with $p,q\geq r$ and ${\mathrm{rank}}(XY)\leq s$ then $\operatorname{dim}\operatorname{Lker}(X)+\operatorname{dim}\operatorname{Rker}Y\geq r-s$ , so in our case we have $(r-{\mathrm{rank}}(X))+(r-\operatorname{rank}(Y))\geq r$ , i.e. ${\mathrm{rank}}(X)+{\mathrm{rank}}(Y)\leq r$ , with $X$ the first matrix and $Y$ the second.

Definition 5.3.

Notations as above. Assume bases have been chosen generically to have the normal form. Write $at_{L}$ for the rank of $At_{L}$ and $at_{R}$ for the rank of $At_{R}$ and call these the Atkinson numbers associated to $E$ .

In particular we have

(7)

at_{L}+at_{R}\leq r.

Say $E\subset B{\mathord{\otimes}}C$ has bounded rank $r$ and we fix $X_{1}=\begin{pmatrix}\operatorname{Id}_{r}&0\\ 0&0\end{pmatrix}$ , i.e., we take a general point of $E$ . $X_{1}:B^{*}\rightarrow C$ defines subspaces $\operatorname{Im}_{\operatorname{Hom}(B^{*},C)}(X_{1})\subset C$ and $\operatorname{ker}_{\operatorname{Hom}(B^{*},C)}(X_{1})^{\perp}=\operatorname{Im}_{\operatorname{Hom}(C^{*},B)}(X_{1})\subset B$ . Call these subspaces $C_{1},B_{1}$ . Let $X\in E$ be another general element. Consider $X:B^{*}\rightarrow C\operatorname{mod}C_{1}$ , and $X:C^{*}\rightarrow B\operatorname{mod}B_{1}$ . The normal form implies that $X(B_{1}^{\perp})\subseteq C_{1}$ and $X(C_{1}^{\perp})\subseteq B_{1}$ The ranks of these maps give two additional invariants. In the coordinate expressions, these are respectively the maps given by $U$ and $W$ . We now allow $X$ to vary, and we ask what is the dimension of the subspaces of $(B_{1}^{\perp})^{*}{\mathord{\otimes}}C_{1}$ and $(C_{1}^{\perp})^{*}{\mathord{\otimes}}B_{1}$ that are filled out? These give two additional invariants, call them $d_{U},d_{W}$ . Let $d_{UW}$ be the dimension of the combined subspaces, in bases, the number of variables appearing in $U,W$ , combined. We know by conciseness (to avoid trivialities) that $(B_{1}^{\perp})^{*}{\mathord{\otimes}}C_{1}$ and $(C_{1}^{\perp})^{*}{\mathord{\otimes}}B_{1}$ respectively surject onto $C_{1},B_{1}$ which lower bounds their dimensions.

Proposition 5.4.

If $r={\mathbb{c}}-1$ and $d_{U}=1$ or $r=\mathbf{b}-1$ and $d_{W}=1$ then $E$ is imprimitive. More generally, for any $\mathbf{b},{\mathbb{c}},r$ , if $U$ can be normalized to be zero except in one column or $W$ can be normalized to be zero except in one row, then $E$ is imprimitive.

Proof.

Say $r=\mathbf{b}-1$ and $d_{W}=1$ . Then in Atkinson normal form we may normalize $W=(w_{1},0,\dotsc,0)^{\mathbb{t}}$ , so

E=\begin{pmatrix}*&w_{1}\\ *&0\\ U&0\end{pmatrix}

where the blocking is $(\mathbf{b}-1,1)\times(1,\mathbf{b}-2,{\mathbb{c}}-\mathbf{b}+1)$ . If we strike out the first row the corank is unchanged. In the general case, the same argument holds when

E=\begin{pmatrix}*&w_{1}&\cdots&w_{k}\\ *&0&\cdots&0\\ U&0&\cdots&0\end{pmatrix}.

∎

Corollary 5.5.

Assume ${\mathbb{a}}\leq\mathbf{b}={\mathbb{c}}$ . A concise tensor $T\in A{\mathord{\otimes}}B{\mathord{\otimes}}C$ such that $T(A^{*})$ is corank one and primitive cannot have minimal border rank $\mathbf{b}$ .

Proof.

By [15, Prop. 3.3], a tensor of minimal border rank as in the hypothesis must have $d_{U}=d_{W}=1$ . ∎

We often slightly abuse notation and let $U,{\mathbb{x}},W$ denote the matrices of linear forms induced by $E$ rather than individual matrices.

Let $E$ be primitive and let $\text{Ann}\,(W)$ denote the submodule of length $r$ row vectors with entries polynomials in the variables appearing in $W$ that annihilate $W$ .

Proposition 5.6.

Let $E\subset B{\mathord{\otimes}}C$ be concise of bounded rank $r$ and primitive. If $\text{Ann}\,(W)$ is generated by row vectors of linear forms and $at_{L}=\operatorname{dim}(\text{Ann}\,(W))=\mathbf{b}-r+1$ , where $\operatorname{dim}(\text{Ann}\,(W))$ is the dimension of the vector space spanned by minimal generators of $\text{Ann}\,(W)$ , then $E$ is expandable.

Proof.

Put $E$ in Atkinson normal $\begin{pmatrix}{\mathbb{x}}&W\\ U&0\end{pmatrix}$ . Write $u_{\alpha}$ for the rows of $U$ . Let

(At_{L})_{prim}=\langle u_{\alpha}{\mathbb{x}}^{j}\mid 1\leq\alpha\leq\mathbf{b}-r,j\geq 0\rangle\subseteq\text{Ann}\,(W).

Let $u\in\langle\textrm{minimal generators of }\text{Ann}\,(W)\rangle\backslash\langle u_{\alpha}\mid 1\leq\alpha\leq\mathbf{b}-r\rangle$ . Then $\widetilde{E}=\begin{pmatrix}{\mathbb{x}}&W\\ U&0\\ u&0\end{pmatrix}$ is also of bounded rank $r$ by Proposition 5.2 as it satisfies the conditions of Atkinson normal form. ∎

5.2. Chern classes and Atkinson numbers

Proposition 5.7.

Notations as in §4.2, then $c_{1}({\mathcal{E}})=at_{R}-c_{1}(\tau_{R})$ and $c_{1}({\mathcal{F}})=at_{L}-c_{1}(\tau_{L})$ , where $\tau_{R},\tau_{L}$ are torsion sheafs that are described explicitly in the proof. In particular, $c_{1}({\mathcal{E}})\leq at_{R}$ and $c_{1}({\mathcal{F}})\leq at_{L}$ .

Proof of Proposition 5.7.

Put the matrix of $\varphi:\underline{B}^{*}\to\underline{C}(1)$ in Atkinson normal form:

\begin{pmatrix}\varphi_{\mathbb{x}}&\varphi_{W}\\ \varphi_{U}&0\end{pmatrix}.

Over the point $[\alpha]\in\mathbb{P}V^{*}$ where $\phi_{[\alpha]}=\begin{pmatrix}\operatorname{Id}_{r}&0\\ 0&0\end{pmatrix}$ , let $B^{*}_{2}\subset B^{*}$ denote $\operatorname{ker}(\phi_{[\alpha]})$ and let $B^{*}_{1}$ be a complement in $B^{*}$ . In other words, $\underline{B}^{*}_{1}$ is the trivial subbundle of $\underline{B}^{*}$ generated by the first $r$ basis vectors. Similarly write $C=C_{1}\oplus C_{2}$ . Note that we may identify $B^{*}_{1}$ and $C_{1}$ .

We analyze $c_{1}({\mathcal{E}})$ . Consider the following maps:

	$\displaystyle\phi_{i}\quad\colon\quad\underline{B}^{}_{2}(-1)\oplus\underline{B}^{}_{2}(-2)\oplus\cdots\oplus\underline{B}^{*}_{2}(-i)$	$\displaystyle\rightarrow\underline{C}_{1}$
	$\displaystyle t_{1}\oplus\cdots\oplus t_{i}$	$\displaystyle\mapsto\varphi_{W}(t_{1})+\varphi_{\mathbb{x}}\varphi_{W}(t_{2})+\cdots+\varphi_{\mathbb{x}}^{i-1}\varphi_{W}(t_{i}).$

Note that $Im(\phi_{i})=Im(\phi_{i+1})$ for all $i\geq r$ . Denote the image of $\phi_{r}$ by ${\mathcal{K}}_{1}$ and cokernel of $\phi_{r}$ by $\mathcal{Q}_{1}$ . Then $\operatorname{rank}{\mathcal{K}}_{1}=at_{R}$ . Explicitly, we may write ${\mathcal{K}}_{1}$ as a subsheaf of $\underline{B}^{*}_{1}$ as follows: Let $\mathcal{V}\subset\mathbb{P}A^{*}$ be an affine open subset containing $[\alpha]$ . Using the identification $B_{1}^{*}\cong C_{1}$ ,

{\mathcal{K}}_{1}\mid_{\mathcal{V}}=\{s\in H^{0}(B^{*}_{1}\mid_{\mathcal{V}})\ \mid\exists t_{i}\in H^{0}(\underline{B}^{*}_{2}(-i)\mid_{\mathcal{V}})\text{\ such\ that\ }s=\varphi_{W}(t_{1})+\varphi_{\mathbb{x}}\varphi_{W}(t_{2})+\ldots+\varphi_{\mathbb{x}}^{r-1}\varphi_{W}(t_{r})\}.

Since for all $k\geq 0$ and all $t\in B_{2}^{*}$ , $\varphi_{U}\varphi_{\mathbb{x}}^{k}\varphi_{W}(t)=0$ , we obtain $\varphi({\mathcal{K}}_{1})\subseteq{\mathcal{K}}_{1}(1)\subset\underline{C}_{1}(1)$ and $\operatorname{rank}\varphi\mid_{{\mathcal{K}}_{1}}=\operatorname{rank}{\mathcal{K}}_{1}=at_{R}$ . We have the following commutative diagram that defines $\varphi^{\mathcal{Q}}$ :

Since $\varphi\mid_{{\mathcal{K}}_{1}}$ is of full rank, $\operatorname{ker}(\varphi\mid_{{\mathcal{K}}_{1}})$ is torsion, but since it is a subsheaf of the torsion free sheaf $\underline{B}_{1}^{*}$ , it is zero, and $\mathrm{coker}(\varphi\mid_{{\mathcal{K}}_{1}})$ is a torsion sheaf. The first Chern classes in the short exact sequence $0\rightarrow{\mathcal{K}}_{1}\rightarrow{\mathcal{K}}_{1}(1)\rightarrow\text{coker}(\varphi\mid_{{\mathcal{K}}_{1}})\rightarrow 0$ show

c_{1}(\text{coker}(\varphi\mid_{{\mathcal{K}}_{1}}))=\operatorname{rank}{\mathcal{K}}_{1}=at_{R}.

Since coker $(\varphi\mid_{{\mathcal{K}}_{1}})$ is torsion, any subsheaf of it has first Chern class no larger than $at_{R}$ .

Claim: $c_{1}(\ker\varphi^{\mathcal{Q}})\geq 0$ .

To see this, since $\varphi(\underline{B}^{*}_{2})=\varphi_{W}(\underline{B}^{*}_{2})\subseteq{\mathcal{K}}_{1}(1)$ , we have $\underline{B}^{*}_{2}\subseteq\ker\varphi^{\mathcal{Q}}$ . The kernel of the map $\varphi^{\mathcal{Q}}\mid_{\mathcal{Q}_{1}}\colon\mathcal{Q}_{1}\rightarrow\mathcal{Q}_{1}(1)\oplus\underline{C}_{2}(1)$ is torsion because $\varphi^{\mathcal{Q}}\mid_{\mathcal{Q}_{1}}$ is of full rank (since $\varphi\mid_{B_{1}^{*}}$ is of full rank). Hence $c_{1}(\ker\varphi^{\mathcal{Q}}\mid_{\mathcal{Q}_{1}})\geq 0$ . Together we have $c_{1}(\ker\varphi^{\mathcal{Q}})\geq 0$ .

Hence we have $c_{1}(\operatorname{Im}(\varphi^{\mathcal{Q}}))=c_{1}(\mathcal{Q}_{1}\oplus\underline{B}^{*}_{2})-c_{1}(\ker\varphi^{\mathcal{Q}})=c_{1}(\mathcal{Q}_{1})-c_{1}(\ker\varphi^{\mathcal{Q}})$ .

We also have the following commutative diagram:

whose rows are exact.

Define ${\mathcal{A}}\colon=\ker(\mathrm{coker}\varphi\mid_{{\mathcal{K}}_{1}}\to\mathrm{coker}\varphi)$ and let ${\mathcal{B}}$ denote the corresponding quotient. We have

0\longrightarrow{\mathcal{A}}\longrightarrow\mathrm{coker}\varphi\mid_{{\mathcal{K}}_{1}}\longrightarrow{\mathcal{B}}\longrightarrow 0.

Recall that $\ker\varphi\mid_{{\mathcal{K}}_{1}}=0$ . Consider the exact sequence

0\longrightarrow\ker\varphi\longrightarrow\ker\varphi^{\mathcal{Q}}\longrightarrow{\mathcal{A}}\longrightarrow 0.

Thus $c_{1}({\mathcal{E}})=-c_{1}(\ker\varphi)=-c_{1}(\ker\varphi^{\mathcal{Q}})+c_{1}({\mathcal{A}})=at_{R}-c_{1}(\operatorname{ker}(\varphi^{\mathcal{Q}}))-c_{1}({\mathcal{B}})$ .

The case of $c_{1}({\mathcal{F}})$ is similar. ∎

Example 5.8.

Let $T(A^{*})\subset B{\mathord{\otimes}}C$ correspond to a maximal dimension $(k_{1},k_{2})$ compression space of rank $r=k_{1}+k_{2}$ so that ${\mathbb{a}}=\mathbf{b}k_{1}+{\mathbb{c}}k_{2}-k_{1}k_{2}$ . A standard way to write such a space is $\begin{pmatrix}*&*\\ *&0\end{pmatrix}$ where the blocking is $(k_{1},\mathbf{b}-k_{1})\times(k_{2},{\mathbb{c}}-k_{2})$ . Permute basis vectors to put it in Atkinson normal form

(8)

\begin{pmatrix}\mathbf{z}_{1}&\mathbf{z}_{2}&\mathbf{z}_{3}\\ 0&\mathbf{z}_{4}&0\\ 0&\mathbf{z}_{5}&0\end{pmatrix}

where the blocking is $(k_{1},k_{2},\mathbf{b}-r)\times(k_{1},k_{2},{\mathbb{c}}-r)$ . Then

\begin{pmatrix}U\\ U{\mathbb{x}}\\ U{\mathbb{x}}^{2}\\ \vdots\\ U{\mathbb{x}}^{r-1}\end{pmatrix}=\begin{pmatrix}(0,\mathbf{z}_{5})\\ (0,\mathbf{z}_{5})\begin{pmatrix}\mathbf{z}_{1}&\mathbf{z}_{2}\\ 0&\mathbf{z}_{3}\end{pmatrix}\\ (0,\mathbf{z}_{5})\begin{pmatrix}\mathbf{z}_{1}&\mathbf{z}_{2}\\ 0&\mathbf{z}_{3}\end{pmatrix}^{2}\\ \vdots\end{pmatrix}=\begin{pmatrix}(0,\mathbf{z}_{5})\\ (0,\mathbf{z}_{5}\mathbf{z}_{3})\\ \vdots\end{pmatrix}

Since the entries of $\mathbf{z}_{5}$ are independent, we obtain $at_{L}=k_{2}$ . Since $\operatorname{det}(\mathbf{z}_{1})$ will factor out of the matrix of size $r$ minors from the first $r$ columns of (8), but the remaining minors are independent because the entries of $\mathbf{z}_{4},\mathbf{z}_{5}$ are independent, we also get $c_{1}({\mathcal{E}})=k_{2}$ . Similarly $at_{R}=c_{1}({\mathcal{F}})=k_{1}$ . If we specialize to a subspace of $\mathbb{P}A^{*}$ , then the Atkinson numbers and Chern classes may drop.

5.3. Chern class defects

Consider

(9)

0\rightarrow{\mathcal{E}}\rightarrow\underline{C}(1)\rightarrow\mathcal{Q}_{{\mathcal{E}}}\rightarrow 0

and the analogous $\mathcal{Q}_{{\mathcal{F}}}$ . We sometimes write $\mathcal{Q}=\mathcal{Q}_{\mathcal{E}}$ in what follows.

Proposition 5.9.

Notations as in §4.2, then $c_{1}({\mathcal{E}})+c_{1}({\mathcal{F}})=r-c_{1}(\mathcal{E}xt^{1}(\mathcal{Q}_{{\mathcal{E}}},\mathcal{O}_{\mathbb{P}A^{*}}))$ . In particular $c_{1}(\mathcal{E}xt^{1}(\mathcal{Q}_{{\mathcal{E}}},\mathcal{O}_{\mathbb{P}A^{*}}))=c_{1}(\mathcal{E}xt^{1}(\mathcal{Q}_{{\mathcal{F}}},\mathcal{O}_{\mathbb{P}A^{*}}))$ .

Note that $c_{1}(\mathcal{E}xt^{1}(\mathcal{Q},\mathcal{O}_{\mathbb{P}A^{*}}))\geq 0$ because it is torsion.

We thank M. Popa for suggesting Proposition 5.9 and an outline of the proof.

Proof of Proposition 5.9.

Dualize (9) to get

(10)

0\rightarrow\mathcal{Q}^{*}\rightarrow\underline{C}^{*}(-1)\rightarrow{\mathcal{E}}^{*}\rightarrow\mathcal{E}xt^{1}(\mathcal{Q},\mathcal{O}_{\mathbb{P}A^{*}}).

Now $\phi^{*}(1):\underline{C}^{*}\rightarrow\underline{B}(1)$ is the map giving rise to ${\mathcal{F}}$ , so $\phi^{*}:\underline{C}^{*}(-1)\rightarrow\underline{B}$ has image ${\mathcal{F}}(-1)$ .

Claim: we also have a map ${\mathcal{E}}^{*}\rightarrow\underline{B}$ . To see this, take a resolution of ${\mathcal{E}}$ by vector bundles

\mathcal{V}_{2}\rightarrow\mathcal{V}_{1}\rightarrow\underline{B}^{*}\rightarrow{\mathcal{E}}\rightarrow 0

and dualize to get

{\mathcal{E}}^{*}\rightarrow\underline{B}\rightarrow\mathcal{V}_{1}^{*}\rightarrow\mathcal{V}_{2}^{*}

which also gives

0\rightarrow\mathcal{Q}^{*}\rightarrow\underline{C}^{*}(-1)\smash{\mathop{\longrightarrow}\limits^{\phi^{*}}}\underline{B}\rightarrow\mathcal{V}_{1}^{*}\rightarrow\mathcal{V}_{2}^{*}

which is not in general exact at $\underline{B}$ . By definition, the homology at that step is ${\mathcal{E}}xt^{1}(\mathcal{Q},{\mathcal{O}}_{\mathbb{P}A^{*}})$ .

By (10), $\phi^{*}$ factors through the map ${\mathcal{E}}^{*}\rightarrow\underline{B}$ . We get an exact sequence

(11)

0\rightarrow{\mathcal{F}}(-1)\rightarrow{\mathcal{E}}^{*}\rightarrow\mathcal{E}xt^{1}(\mathcal{Q},\mathcal{O}_{\mathbb{P}A^{*}})\rightarrow 0

Thus $c_{1}({\mathcal{E}}^{*})=c_{1}({\mathcal{F}})-r+c_{1}(\mathcal{E}xt^{1}(\mathcal{Q},\mathcal{O}_{\mathbb{P}A^{*}}))$ , i.e.,

c_{1}({\mathcal{E}})+c_{1}({\mathcal{F}})=r-c_{1}(\mathcal{E}xt^{1}(\mathcal{Q},\mathcal{O}_{\mathbb{P}A^{*}})).

∎

Remark 5.10.

Dualizing (11), we see the error in the assertion about ${\mathcal{E}}^{**}$ and ${\mathcal{F}}^{*}(1)$ is measured by

0\rightarrow{\mathcal{E}}^{**}\rightarrow{\mathcal{F}}^{*}(1)\rightarrow\mathcal{E}xt^{1}(\mathcal{E}xt^{1}(\mathcal{Q},\mathcal{O}_{\mathbb{P}A^{*}}),\mathcal{O}_{\mathbb{P}A^{*}}))\rightarrow\mathcal{E}xt^{1}({\mathcal{E}}^{*},\mathcal{O}_{\mathbb{P}A^{*}})\rightarrow\ldots.

Example 5.11.

Consider the following spaces from [15], which, as tensors form a complete list of concise, $1$ -degenerate tensors in $(\mathbb{C}^{5})^{{\mathord{\otimes}}3}$ of minimal border rank:

Represented as spaces of matrices, the tensors may be presented as:

	$\displaystyle T_{{\mathcal{O}}_{58}}$	$\displaystyle=\begin{pmatrix}x_{1}&&x_{2}&x_{3}&x_{5}\\ x_{5}&x_{1}&x_{4}&-x_{2}&\\ &&x_{1}&&\\ &&-x_{5}&x_{1}&\\ &&&x_{5}&\end{pmatrix},\ \ T_{{\mathcal{O}}_{57}}=\begin{pmatrix}x_{1}&&x_{2}&x_{3}&x_{5}\\ &x_{1}&x_{4}&-x_{2}&\\ &&x_{1}&&\\ &&&x_{1}&\\ &&&x_{5}&\end{pmatrix},$
	$\displaystyle T_{{\mathcal{O}}_{56}}$	$\displaystyle=\begin{pmatrix}x_{1}&&x_{2}&x_{3}&x_{5}\\ &x_{1}+x_{5}&&x_{4}&\\ &&x_{1}&&\\ &&&x_{1}&\\ &&&x_{5}&\end{pmatrix},\ \ T_{{\mathcal{O}}_{55}}=\begin{pmatrix}x_{1}&&x_{2}&x_{3}&x_{5}\\ &x_{1}&x_{5}&x_{4}&\\ &&x_{1}&&\\ &&&x_{1}&\\ &&&x_{5}&\end{pmatrix},\ \ T_{{\mathcal{O}}_{54}}=\begin{pmatrix}x_{1}&&x_{2}&x_{3}&x_{5}\\ &x_{1}&&x_{4}&\\ &&x_{1}&&\\ &&&x_{1}&\\ &&&x_{5}&\end{pmatrix}.$

These are all bounded rank four. The Chern classes $(c_{1}({\mathcal{E}}),c_{1}({\mathcal{F}}))$ equal the Atkinson numbers and are respectively $(2,2),(1,1),(1,1),(1,1),(1,1)$ . This is most easily computed via the Chern classes of the kernel bundles.

These are all compression spaces, as permuting the third and fifth columns makes the lower $3\times 3$ block zero. They are degenerations of the general such compression space over $\mathbb{P}^{15}$ where the kernel sheaves are isomorphic and move in a $Seg(\mathbb{P}^{1}\times\mathbb{P}^{2})\subset\mathbb{P}^{5}$ , which means ${\mathcal{K}}={\mathcal{O}}_{\mathbb{P}^{15}}(-2)$ . In particular the Chern classes can decrease under specialization.

Note in the $(1,1)$ cases we have $c_{1}({\mathcal{E}})=1$ , $c_{1}({\mathcal{E}}^{**})=c_{1}({\mathcal{F}}^{*}(1))=4-1=3$ , so the Eisenbud-Harris assertions about ${\mathcal{E}}$ and ${\mathcal{F}}^{*}(1)$ fail in these cases.

6. Proof of the Main Theorem

Outline of the proof: In §6.1, we make general remarks on the $c_{1}({\mathcal{E}})=2$ corank one case. The $(5,5)$ case is treated in §6.2. The $(5,m)$ cases, with $m>7$ are ruled out by [2, Thm. B]. We refine the proof of [2, Thm. B] to rule out the $(5,7)$ case in §6.3. We reformulate the Kronecker normal form for pencils of matrices and use it to analyze linear annihilators of spaces of linear forms of $2\times r$ matrices in §6.4. In §6.5 we use our analysis in §6.4 to reduce to two new potentially basic spaces. The $(6,5)$ case is treated in §6.6. In §6.7 we go to a slightly more general setting and observe the Chern classes in the two $(6,6)$ examples are indeed all $2$ . In §6.8 we prove the two $(6,6)$ examples are indeed basic.

6.1. Zero sets and syzygies of spaces of quadrics

Continuing the notation of §4.2, write

0\rightarrow{\mathcal{K}}_{\mathcal{E}}\rightarrow\underline{B}^{*}\rightarrow C(1),

and

0\rightarrow{\mathcal{K}}_{\mathcal{F}}\rightarrow\underline{C}^{*}\rightarrow B(1),

which we dualize and twist to get

\underline{B}^{*}\rightarrow C(1)\rightarrow{\mathcal{K}}_{\mathcal{F}}^{*}(1).

Since the maps $B^{*}\rightarrow C(1)$ agree we obtain

(12)

0\rightarrow{\mathcal{K}}_{\mathcal{E}}\rightarrow\underline{B}^{*}\rightarrow C(1)\rightarrow{\mathcal{K}}_{\mathcal{F}}^{*}(1),

which is a complex but not in general exact. Write the maps as $d_{3},d_{2},d_{1}$ respectively.

Recall the dictionary between sheaves on projective space and graded modules. Write $R=\mathbb{C}[A]$ for the ring of polynomials on $A^{*}$ and adopt the usual convention that $R(k)$ is $R$ with the grading shifted by $k$ . In terms of modules, assuming $c_{1}({\mathcal{E}})=c_{1}({\mathcal{F}})=2$ , the corank one $(m,m)$ case becomes:

Here the entries of the vector $\operatorname{ker}\varphi$ are quadrics, which, by hypothesis have no common factor, and similarly for the entries of $\operatorname{ker}\varphi^{\mathbb{t}}$ .

In this section we reduce the corank one case with $c_{1}({\mathcal{E}})=2$ . We already saw that the kernel map is given by a row vector of quadrics which have no common factor. We then specialize to the basic rank four case and show the zero set of the quadrics always has codimension at least three.

For $B\subset S^{2}A^{*}$ , let $K_{21}(B)$ be the space of linear syzygies, i.e., the kernel of the multiplication map $B{\mathord{\otimes}}A^{*}\rightarrow S^{3}A^{*}$ . We will be interested in the corank one space of matrices given by the tensor inducing $A^{*}\subset B{\mathord{\otimes}}K_{21}(B)^{*}=:B{\mathord{\otimes}}C$ .

Let $B\subset S^{2}A^{*}$ , let $\mathbf{b}=\operatorname{dim}B\geq 3$ (the case of $\mathbf{b}=2$ is handled by Kronecker normal form).

Let $Q_{1},Q_{2}\in B$ be general and consider their zero set. Then either

(1)

The zero set is a degree four codimension two irreducible complete intersection. In this case intersecting with a third general quadric in $B$ will provide a codimension three set.
(2)

The zero set is of pure codimension two, consisting of the union of a cone over the twisted cubic $v_{3}(\mathbb{P}^{1})$ and a linear space.
(3)

The zero set is codimension two and the codimension two component consists of two irreducible quadrics, each in a hyperplane. In this case the space spanned by $Q_{1},Q_{2}$ is $\langle Q,\ell m\rangle$ , where $Q$ is irreducible and $\ell,m\in A^{*}$ , and the two irreducible quadrics are ${\rm Zeros}(Q,\ell)$ , ${\rm Zeros}(Q,m)$ . Then, by the genericity hypothesis, $B=\langle Q,\ell F\rangle$ where $\operatorname{dim}F=\mathbf{b}-1$ and $\operatorname{dim}K_{21}(B)=\binom{\mathbf{b}-1}{2}$ .
(4)

The zero set is codimension two and the codimension two components consist of linear spaces. Then $B=\langle\ell_{1}F_{1},\ell_{2}F_{2}\rangle$ , for some linear subspaces $F_{1},F_{2}\subset A^{*}$ .
(5)

The zero set has codimension one: then $B=\ell F$ where $\ell\in A^{*}$ and $F\subset A^{*}$ and then $\operatorname{dim}K_{21}(B)=\binom{\mathbf{b}}{2}$ .

The only case requiring explanation is (2), which follows from the classification of varieties of minimal degree, see, e.g., [11], which in particular says that a degree three irreducible variety of codimension two is a cone over the twisted cubic.

We now consider what possible basic spaces of bounded rank $\mathbf{b}-1$ in $B{\mathord{\otimes}}C=B{\mathord{\otimes}}K_{21}(B)$ could arise with kernel of dimension one given by the quadrics in $B$ . We assume $\mathbf{b}>4$ . We claim only case (1) can potentially occur: Case (2) cannot occur because in this case $\operatorname{dim}K_{21}(B)=2$ . The remaining cases are ruled out because in each case there will be vectors of linear forms in the kernel.

6.2. Conclusion of the $(5,5)$ case

Lemma 6.1.

The only basic space of $5\times 5$ matrices of linear forms with $c_{1}({\mathcal{E}})=c_{1}({\mathcal{F}})=2$ is the space of skew-symmetric matrices.

Proof.

Let $I$ denote the ideal generated by the quadrics in $\operatorname{ker}\varphi$ . By the discussion in §6.1, $\mathrm{depth}\;I=\mathrm{codim}\;I\geq 3$ .

Set $I_{k}(\varphi)$ to be the ideal generated by the size $k$ minors of $\phi$ . Using that $\langle\Lambda^{4}\phi\rangle=(\operatorname{ker}\phi)(\operatorname{ker}\phi^{\mathbb{t}})$ , we have $\mathrm{gcd}(I_{4}(\varphi))=1$ as $c_{1}({\mathcal{E}})=c_{1}({\mathcal{F}})=2$ . This implies $\mathrm{depth}\;I_{4}(\varphi)\geq 2$ and $\mathrm{depth}\;I_{1}(\ker\varphi)\geq 3$ . This is enough for $\mathrm{coker}\;\varphi$ to be a first syzygy module, so that we can complete the resolution $\mathbb{F}_{\bullet}$ on the left:

By assumption, since $\varphi$ is primitive, the entries in $\ker\varphi^{\mathbb{t}}$ are linearly independent. We are forced to have $\mathrm{depth}\;I_{1}(\ker\varphi^{\mathbb{t}})=\mathrm{codim}\;I_{1}(\ker\varphi^{\mathbb{t}})\geq 3$ . By the Buchsbaum-Eisenbud criterion for exactness [5], we conclude that both $\mathbb{F}_{\bullet}$ and its dual $\mathbb{F}_{\bullet}^{*}$ are exact. Hence $\mathbb{F}_{\bullet}$ is self dual (up to shift) and it defines a Gorenstein ideal of depth $3$ . By the characterization of Gorenstein ideals of depth 3 [6, Thm. 2.1(2)], $\varphi$ is skew-symmetrizable. ∎

6.3. $(r+1)\times m$ spaces of bounded rank $r$

A basis of the linear annihilator of a vector of linear forms $(a_{1},\dotsc,a_{k})$ is

\begin{pmatrix}-a_{2}\\ a_{1}\\ 0\\ \vdots\\ \vdots\end{pmatrix},\begin{pmatrix}-a_{2}\\ 0\\ a_{1}\\ 0\\ \vdots\end{pmatrix},\ldots,\begin{pmatrix}-a_{k}\\ 0\\ \vdots\\ 0\\ a_{1}\end{pmatrix},\begin{pmatrix}0\\ -a_{3}\\ a_{2}\\ 0\\ \vdots\\ \vdots\end{pmatrix},\begin{pmatrix}0\\ -a_{4}\\ 0\\ a_{2}\\ 0\\ \vdots\end{pmatrix},\ldots,\begin{pmatrix}0\\ -a_{k}\\ 0\\ \vdots\\ 0\\ a_{2}\end{pmatrix},\ldots,\begin{pmatrix}0\\ \vdots\\ 0\\ -a_{k}\\ a_{k-1}\end{pmatrix}.

In particular, it has dimension $\binom{k}{2}$ .

From this, one may recover the Atkinson result:

Theorem 6.2.

[2, Thm. B] Let $E\subset\mathbb{C}^{r+1}{\mathord{\otimes}}\mathbb{C}^{m}$ be a primitive space of bounded rank $r$ and let $m>r+\binom{r-1}{2}$ . Then $E$ is a specialization of $\mathbb{C}^{r+1}\subset\operatorname{Hom}(\mathbb{C}^{r+1},\Lambda^{2}\mathbb{C}^{r+1})$ .

The idea of the proof is that assuming $U$ has no entries equal to zero, there are enough elements of the linear annihilator of $U$ present so the $U{\mathbb{x}}W=0$ equation forces ${\mathbb{x}}$ to be a scalar times the identity. Writing out the matrix of linear forms of $V\rightarrow\operatorname{Hom}(V,\Lambda^{2}V)$ one sees this implies the space is a specialization of $V\rightarrow\operatorname{Hom}(V,\Lambda^{2}V)$ .

We now consider rank $4$ spaces in $\mathbb{C}^{5}{\mathord{\otimes}}\mathbb{C}^{7}$ :

First assume all entries of $U=(u_{1},u_{2},u_{3},u_{4})$ are linearly independent. In order to avoid ${\mathbb{x}}$ being forced to be a linear form times the identity by the $U{\mathbb{x}}W=0$ equation, $W$ is uniquely determined up to isomorphism, namely

W=\begin{pmatrix}-u_{2}&-u_{3}&-u_{4}\\ u_{1}&&\\ &u_{1}&\\ &&u_{1}\end{pmatrix}.

Here $u_{1}^{2}$ divides all size three minors, and one would obtain $c_{1}({\mathcal{F}})=1$ , so this case is eliminated.

When $U$ is three dimensional, normalize so $U=(0,u_{2},u_{3},u_{4})$ . Then if the first row of $W$ is zero, the full annihilator of $(u_{2},u_{3},u_{4})$ must appear in $W$ . Then using the $U{\mathbb{x}}W=0$ equation we see that the first row of ${\mathbb{x}}$ is zero except in the $(1,1)$ slot. Otherwise the first row of $W$ contains a nonzero entry and $U{\mathbb{x}}W=0$ gives the same conclusion for the first column of ${\mathbb{x}}$ . Striking the first row (resp. column) gives a space of bounded rank $r-1$ so the space is not primitive.

When $U$ is two-dimensional, write $U=(0,0,u_{3},u_{4})$ , so we must have, after a change of basis

W=\begin{pmatrix}w^{1}_{1}&w^{1}_{2}&w^{1}_{3}\\ w^{2}_{1}&w^{2}_{2}&w^{2}_{3}\\ &&-u_{4}\epsilon\\ &&u_{3}\epsilon\end{pmatrix}.

with the upper left $2\times 2$ block of full rank to avoid a column of zeros after a change of basis and $\epsilon\in\{0,1\}$ . Then $U{\mathbb{x}}W=0$ implies the lower left $2\times 2$ block of ${\mathbb{x}}$ is zero. Combining this with the first two columns of $W$ and below, we obtain a $4\times 3$ block of zeros and the space is compression.

When $U$ is one-dimensional, the space is imprimitive by Proposition 5.4.

We conclude:

Proposition 6.3.

Let $E\subset\mathbb{C}^{5}{\mathord{\otimes}}\mathbb{C}^{7}$ be a primitive space of bounded rank $4$ . Then $E$ is a specialization of $\mathbb{C}^{5}\subset\operatorname{Hom}(\mathbb{C}^{5},\Lambda^{2}\mathbb{C}^{5})$ .

6.4. $2\times k$ spaces of linear forms and their linear annihilators

Recall the Kronecker normal form for pencils of matrices, i.e., tensors in $A{\mathord{\otimes}}B{\mathord{\otimes}}F$ with $\operatorname{dim}F=2$ : all are block matrices with blocks

L_{k}(F^{*}):=\begin{pmatrix}s&t&0&\cdots&0\\ 0&s&t&0&\cdots&\\ &&\ddots&&\\ &&&s&t\end{pmatrix},\ L_{k}^{\mathbb{t}}(F^{*})=\begin{pmatrix}s&0&0&\cdots&0\\ t&s&0&0&\cdots&\\ &&\ddots&&\\ &&\ddots&&\\ &&&t&s\end{pmatrix},\ Jor_{k,\lambda}(F^{*})=s\operatorname{Id}_{k}+tJ

where the matrices are respectively $(k+1)\times k$ , $k\times(k+1)$ , $k\times k$ and in the last case $J$ is a single Jordan block with eigenvalue $\lambda$ . Of particular note is $L_{1}(F^{*})=(s,t)$ and $L_{1}^{\mathbb{t}}(F^{*})=\begin{pmatrix}s\\ t\end{pmatrix}$ . We adopt the convention that the rows are indexed by $A$ and the columns by $B$ . Rewritten as a linear subspace of $B{\mathord{\otimes}}F$ these become:

L_{k}(A^{*})=\begin{pmatrix}a_{1}&a_{2}&\cdots&a_{k}&0\\ 0&a_{1}&&\cdots&a_{k}\end{pmatrix},\ L_{k}^{\mathbb{t}}(A^{*})=\begin{pmatrix}a_{1}&a_{2}&\cdots&a_{k}\\ a_{2}&a_{3}&\cdots&a_{k+1}\end{pmatrix},

Jor_{k,\lambda}(A^{*})=\begin{pmatrix}a_{1}&a_{2}&\cdots&a_{k}\\ \lambda a_{1}&\lambda a_{2}+a_{1}&\cdots&\lambda a_{k}+a_{k-1}\end{pmatrix}.

Note the special cases

L_{1}(A^{*})=\begin{pmatrix}a&0\\ 0&a\end{pmatrix},\ L_{1}^{\mathbb{t}}(A^{*})=\begin{pmatrix}a_{1}\\ a_{2}\end{pmatrix},\ Jor_{1,0}(A^{*})=\begin{pmatrix}a\\ 0\end{pmatrix},\ Jor_{1,\lambda}(A^{*})=\begin{pmatrix}a\\ \lambda a\end{pmatrix},\ Jor_{2,0}=\begin{pmatrix}a_{1}&a_{2}\\ 0&a_{1}\end{pmatrix}.

Let $0\leq u\leq v\leq f$ , Write $A=A_{1}\oplus\cdots\oplus A_{f}$ . The general form is,

(L_{k_{1}}(A_{1}^{*}),\dotsc,L_{k_{u}}(A_{u}^{*}),L_{\ell_{1}}(A_{u+1}^{*}),\dotsc,L_{\ell_{v}}(A_{v}^{*}),Jor_{i_{1},\lambda_{i_{1}}}(A_{v+1}^{*}),\dotsc,Jor_{i_{f},\lambda_{i_{f}}}(A_{f}^{*})).

Now we study linear annihilators. First note that if a space consists of a single block, it has no linear annihilator. We next consider pairs of blocks. Label the linear forms in the first block with $a_{j}$ ’s and those in the second with $b_{j}$ ’s.

An $L_{k}$ and an $L_{q}$ , or an $L_{k}$ and a $J_{q,0}$ with $q>1$ , or a $J_{k,0}$ , $k>1$ and a $J_{q,0}$ , $q>1$ : one gets a two dimensional space. Respectively,

\begin{pmatrix}-b_{1}\\ 0\\ \vdots\\ 0\\ a_{1}\\ 0\\ \vdots\\ 0\end{pmatrix},\begin{pmatrix}0\\ \vdots\\ 0\\ -b_{q}\\ 0\\ \vdots\\ 0\\ a_{q}\end{pmatrix},\ \ {\rm and}\ \ \begin{pmatrix}-b_{1}\\ 0\\ \vdots\\ 0\\ a_{1}\\ 0\\ \vdots\\ 0\end{pmatrix},\begin{pmatrix}-b_{2}\\ -b_{1}\\ 0\\ \vdots\\ a_{1}\\ a_{2}\\ 0\\ \vdots\end{pmatrix},\ \ {\rm and}\ \ \begin{pmatrix}-b_{1}\\ 0\\ 0\\ \vdots\\ 0\\ a_{1}\\ 0\\ 0\\ \vdots\\ 0\end{pmatrix},\begin{pmatrix}-b_{2}\\ -b_{1}\\ 0\\ \vdots\\ 0\\ a_{2}\\ a_{1}\\ 0\\ \vdots\\ 0\end{pmatrix}.

An $L_{k}$ with a $J_{1,0}$ has a one-dimensional linear annihilator. A $J_{k,0}$ (any $k$ ) with a $J_{1,0}$ also has a one-dimensional linear annihilator.

An $L_{1}$ and an $L_{q}^{\mathbb{t}}$ or a $Jor_{q,\lambda}$ : one gets a $q$ dimensional space. Respectively:

\begin{pmatrix}-b_{1}\\ -b_{2}\\ 0\\ \vdots\\ 0\\ a_{1}\\ 0\\ \vdots\\ 0\end{pmatrix},\dotsc,\begin{pmatrix}-b_{q}\\ -b_{q+1}\\ 0\\ \vdots\\ 0\\ 0\\ 0\\ \vdots\\ a_{1}\end{pmatrix},\ \ {\rm and}\ \ \begin{pmatrix}-b_{1}\\ -\lambda b_{1}\\ 0\\ \vdots\\ 0\\ a_{1}\\ 0\\ \vdots\\ 0\end{pmatrix},\begin{pmatrix}-b_{2}\\ -\lambda b_{2}-b_{3}\\ 0\\ \vdots\\ 0\\ 0\\ a_{1}\\ 0\\ \vdots\end{pmatrix},\dotsc,\begin{pmatrix}-b_{q}\\ -\lambda b_{q}-b_{q+1}\\ 0\\ \vdots\\ 0\\ 0\\ 0\\ \vdots\\ a_{1}\end{pmatrix}.

All other pairs have no linear annihilator.

6.5. $(6,6)$ -spaces

We now specialize to spaces of $2\times 4$ matrices with linear forms as entries with at least a two-dimensional linear annihilator. The discussion in §6.4 implies the linear annihilator is at most two-dimensional.

The same argument as in the $(5,7)$ case shows that $U$ cannot have any columns equal to zero.

Going through partitions of four, we already know we need at least two parts.

The case of $(3,1)$ is ruled out by our general analysis above.

We consider the $(2,2)$ and $(2,1,1)$ cases together. Thus we have $U=(u_{1},u_{2})$ , where the variables appearing in $u_{1}$ are independent of those in $u_{2}$ and each has two columns, then $W=\begin{pmatrix}-u_{2}\\ u_{1}\end{pmatrix}$ . By our analysis above, in order to have a two dimensional linear annihilator, either $u_{1}=L_{1}=\mu\operatorname{Id}$ and $u_{2}$ can be anything, or $u_{1}$ and $u_{2}$ are $J_{2,0}$ ’s. Note that in either case, $u_{1},u_{2}$ must commute. We have the following possibilities:

$u_{1}=L_{1}$ and $u_{2}$ anything, i.e., one of $L_{1},J_{2,0},J_{2,\lambda},L_{1}^{\mathbb{t}}L_{1}^{\mathbb{t}},L_{1}^{\mathbb{t}}J_{1,\lambda},J_{1,\lambda}J_{1,\lambda^{\prime}},J_{1,\lambda}J_{1,0}L_{1}^{\mathbb{t}}J_{1,0},J_{1,0}J_{1,0}$ where $\lambda,\lambda^{\prime}\neq 0$ , or

$u_{1}=J_{2,0}$ and $u_{2}=J_{2,0}$ .

Note the only cases with $u_{2}$ non-invertible are $u_{2}=J_{1,0}J_{1,0}$ and $u_{2}=J_{1,\lambda}J_{1,\lambda}$ (same $\lambda$ in both).

Write ${\mathbb{x}}=\begin{pmatrix}x_{1}&x_{2}\\ x_{3}&x_{4}\end{pmatrix}$ with each $x_{j}$ $2\times 2$ , then the $U{\mathbb{x}}W=0$ equation gives the equation of $2\times 2$ matrices with cubic entries:

-u_{1}x_{1}u_{2}-u_{2}x_{3}u_{2}+u_{1}x_{2}u_{1}+u_{2}x_{4}u_{1}=0

Write $x_{j}=y_{j}+x_{j}(u_{1})+x_{j}(u_{2})$ where the variables in $y_{j}$ are independent of those in $u_{1},u_{2}$ and $x_{j}(u_{i})$ has entries linear in the entries of $u_{i}$ . We immediately obtain $y_{2}=x_{2}(u_{1})=0$ and if $u_{2}\neq J_{1,0}J_{1,0}$ , $y_{3}=x_{3}(u_{2})=0$ .

Note we are free to modify $x_{1}$ (resp. $x_{3}$ ) by a multiple of $u_{1}$ as long as we modify $x_{2}$ (resp. $x_{4})$ , by the same multiple of $u_{2}$ , and $x_{1}$ (resp. $x_{2}$ ) by a multiple of $-u_{2}$ as long as we modify $x_{3}$ (resp. $x_{4}$ ) by the same multiple of $u_{1}$ .

Consider the case $u_{1}=u_{2}=J_{2,0}$ . Note that the centralizer (among matrices of linear forms) of a $J_{2,0}$ is spanned by $t\operatorname{Id}$ and $J_{2,0}$ . Separating the equations by the variables involved, we obtain $y_{1}=y_{4}=t\operatorname{Id}+J_{2,0}$ , and we may absorb the $t\operatorname{Id}$ into the $J_{2,0}$ . Taking into account the other homogeneities and our allowed modifications, we obtain Case (IV).

Next consider the case $u_{1}=L_{1}$ and $u_{2}$ has centralizer $t\operatorname{Id}$ , which occurs when $u_{2}$ is any of $J_{2,\lambda},L_{1}^{\mathbb{t}}L_{1}^{\mathbb{t}},L_{1}^{\mathbb{t}}J_{1,\lambda},L_{1}^{\mathbb{t}}J_{1,0}$ . In all these cases we are reduced to ${\mathbb{x}}=t\operatorname{Id}_{4}$ , but the resulting spaces are all specializations of the case $u_{2}=L_{1}^{\mathbb{t}}L_{1}^{\mathbb{t}}$ , and thus not basic unless $u_{2}=L_{1}^{\mathbb{t}}L_{1}^{\mathbb{t}}$ , which is Case (III).

Case $(u_{1},u_{2})=(L_{1},L_{1})$ : the solutions to the homogeneous parts of the equations, after admissible modifications, give $y_{1}=y_{4}$ , and all other terms in ${\mathbb{x}}$ are zero. The resulting space is a permuted version of Case (III).

Case $u_{1}=L_{1}$ , $u_{2}=J_{2,0}$ . This case gives a specialization of Case (IV).

Case $u_{1}=L_{1}$ , $u_{2}=J_{1,0}J_{1,0}$ . The $U{\mathbb{x}}W=0$ equation implies $x^{2}_{2}$ is the only nonzero entry in the second row, so this case is not primitive.

The case $(1,1,1,1)$ is easily ruled out.

6.6. $(6,5)$ case

The same argument as in the $(5,7)$ case shows that $U$ cannot have any columns equal to zero. Thanks to Proposition 5.6, the cases that might give basic spaces are when $U$ has a one-dimensional linear annihilator and $at_{R}=2$ , so ${\mathbb{x}}W$ must not be a linear form times $W$ , and thus $U$ has a primitive degree two annihilator. By the discussion in §6.4, there are four cases with a one-dimensional linear annihilator: $(L_{2},J_{1,0})$ , $(J_{3,0},J_{1,0})$ , $(J_{2,0},J_{1,0},L_{1}^{\mathbb{t}})$ , and $(J_{1,0},J_{1,0},L_{1}^{\mathbb{t}},L_{1}^{\mathbb{t}})$ which we may respectively write as

\begin{pmatrix}u_{1}&u_{2}&0&u_{3}\\ 0&u_{1}&u_{2}&0\end{pmatrix},\ \begin{pmatrix}u_{1}&u_{2}&u_{3}&u_{4}\\ 0&u_{1}&u_{2}&0\end{pmatrix},\ \begin{pmatrix}u_{1}&u_{2}&u_{3}&u_{4}\\ 0&u_{1}&0&u_{5}\end{pmatrix},\ \begin{pmatrix}u_{1}&u_{2}&u_{3}&u_{4}\\ 0&0&u_{5}&u_{6}\end{pmatrix}.

Note that the first is a degeneration of the second, and after permuting columns, the second is a degeneration of the third, and the third a degeneration of the fourth. To make the degeneration transparent, we rename the variables and write the spaces as follows

\begin{pmatrix}u_{1}&u_{2}&u_{3}&0\\ 0&0&u_{1}&u_{3}\end{pmatrix},\ \begin{pmatrix}u_{1}&u_{2}&u_{3}&u_{4}\\ 0&0&u_{1}&u_{3}\end{pmatrix},\ \begin{pmatrix}u_{1}&u_{2}&u_{3}&u_{4}\\ 0&0&u_{1}&u_{6}\end{pmatrix},\ \begin{pmatrix}u_{1}&u_{2}&u_{3}&u_{4}\\ 0&0&u_{5}&u_{6}\end{pmatrix}.

In all the cases, $W=(-u_{2},u_{1},0,0)^{\mathbb{t}}$ . We computed by Macaulay2 that primitive degree two annihilator of $U$ is generated by $(0,-u_{4}u_{5}+u_{3}u_{6},-u_{2}u_{6},u_{2}u_{5})^{\mathbb{t}}$ and $(-u_{4}u_{5}+u_{3}u_{6},0,-u_{1}u_{6},u_{1}u_{5})^{\mathbb{t}}$ , where the effect of degeneration does not change the space of primitive degree two annihilators. By analyzing the first entry of ${\mathbb{x}}W$ , we see that ${\mathbb{x}}W$ has to be a linear form times $W$ since it cannot involve any primitive degree two annihilators, which is a contradiction.

6.7. Good Chern classes

By construction the Atkinson numbers in our examples are all $2$ , here we see the Chern classes are as well. We work in a slightly more general setting where $u_{1},u_{2},x_{1}$ are commuting $k\times k$ matrices, i.e., the general setting of Proposition 2.3 applied to $\mathbb{C}^{3}\subset\operatorname{Hom}(\mathbb{C}^{3},\Lambda^{2}\mathbb{C}^{3})$ . Our space is

\begin{pmatrix}x_{1}&&-u_{2}\\ &x_{1}&u_{1}\\ u_{1}&u_{2}&\end{pmatrix}

Then the left kernel is the $k\times 3k$ matrix of linear forms $(-u_{1},-u_{2},x_{1})$ , and the right kernel is the $3k\times k$ matrix $\begin{pmatrix}u_{2}\\ -u_{1}\\ x_{1}\end{pmatrix}$ . Since the blocks $u_{1},u_{2},x_{1}$ are each in different sets of variables, the matrix of size $k$ minors has no common factor and we conclude $c_{1}({\mathcal{E}})=c_{1}({\mathcal{F}})=k$ .

6.8. Proof that our two new examples are basic

Proof.

By the Buchsbaum-Eisenbud criterion for exactness [5], it is easy to see that the following complexes are exact as well as their duals:

where $R=\mathbb{C}[a_{1},\ldots,a_{6}]$ ,

d_{3}=\left(\begin{smallmatrix}a_{3}&a_{5}\\ a_{4}&a_{6}\\ -a_{2}&0\\ 0&a_{2}\\ a_{1}&0\\ 0&a_{1}\end{smallmatrix}\right),\ d_{1}=\left(\begin{smallmatrix}-a_{2}&0&-a_{3}&-a_{5}&a_{1}&0\\ 0&-a_{2}&-a_{4}&-a_{6}&0&a_{1}\end{smallmatrix}\right),\ d_{3}^{{}^{\prime}}=\left(\begin{smallmatrix}a_{5}&a_{6}\\ 0&a_{5}\\ -a_{3}&-a_{4}\\ 0&-a_{3}\\ a_{1}&a_{2}\\ 0&a_{1}\end{smallmatrix}\right),{\rm\ and\ }d_{1}^{{}^{\prime}}=\left(\begin{smallmatrix}-a_{3}&-a_{4}&-a_{5}&-a_{6}&a_{1}&a_{2}\\ 0&-a_{3}&0&-a_{5}&0&a_{1}\end{smallmatrix}\right).

This proves Case (III) and Case (IV) are unexpandable.

The spaces Case (III) and Case (IV) are strongly indecomposable if and only if the image sheaves of $d_{1},d_{1}^{{}^{\prime}}$ as well as the image sheaf of $d_{3}^{\mathbb{t}},(d_{3}^{{}^{\prime}})^{\mathbb{t}}$ are indecomposable. In the case corresponding to Case (III), we compute the ideal generated by maximal minors of $d_{3}$ as well as $d_{1}$ . They each have $12$ minimal generators. However, if their image sheaf were decomposable, the number of minimal generators would be at most $\binom{5}{2}=10$ . To see this, one obtains the most generators when the last column of $d_{3}^{\mathbb{t}}$ or $d_{1}$ is zero. In the case corresponding to Case (IV), note that the entries of the first row vector of $d_{1}^{{}^{\prime}}$ as well as the entries of the second column vector of $d_{3}^{{}^{\prime}}$ are algebraically independent (even after any possible row operations). Hence up to row/column operations and $\mathrm{GL}(A)$ actions, we cannot have a zero entry in the first row of $d_{1}^{{}^{\prime}}$ (respectively, the second column of $d_{3}^{{}^{\prime}}$ ). Thus the image sheaves of $d_{1}^{{}^{\prime}}$ and $d_{3}^{{}^{\prime}}$ are indecomposable.

Finally, to show Case (III) and Case (IV) are unliftable, we used code in Macaulay2 implementing Proposition 4.3. ∎

7. Additional Examples

We first give two corank two examples generalizing the new rank four cases: If there are exactly two $L_{1}$ blocks and at most one $Jor_{1,0}$ block, then there is a $2\mathbf{b}-6$ dimensional kernel, and the resulting space of bounded rank $\mathbf{b}-2$ size $\mathbf{b}\times(2\mathbf{b}-6)$ matrices is a specialization of the case with $\mathbf{b}-4$ $L_{1}^{\mathbb{t}}$ ’s, so it is sufficient consider that space. That is, set $p=\mathbf{b}-3$ ,the space is

(13)

\begin{pmatrix}a_{1}&0&a_{2}&0&a_{3}&a_{5}&\cdots&a_{2p-1}\\ 0&a_{1}&0&a_{2}&a_{4}&a_{6}&\cdots&a_{2p}\end{pmatrix}

and after permuting rows of the kernel matrix to make it in Atkinson normal form one obtains

(14)

\begin{pmatrix}a_{1}&&&&&&-a_{3}&-a_{5}&\cdots&-a_{2p-1}\\ &a_{1}&&&&&-a_{4}&-a_{6}&\cdots&-a_{2p}\\ &&\ddots&&&&a_{2}&&&\\ &&&a_{1}&&&&a_{2}&&\\ &&&&a_{1}&&&&\ddots&\\ &&&&&a_{1}&&&&a_{2}\\ -a_{2}&0&-a_{3}&-a_{5}&\cdots&-a_{2p-1}&0&\cdots&&0\\ 0&-a_{2}&-a_{4}&-a_{6}&\cdots&-a_{2p}&0&\cdots&&0\end{pmatrix}

This is a $2(\mathbf{b}-3)$ -dimensional space of $\mathbf{b}\times 2\mathbf{b}-6$ matrices of bounded rank $\mathbf{b}-2$ generalizing Case (III).

The example Case (IV) generalizes to a $3q$ -dimensional space of $[2q+\binom{q}{2}]\times[2q+2]$ matrices of bounded rank $2q$ .

More generally, going beyond corank two using the same blow-up, with $X$ a $k\times k$ matrix of linear forms

\begin{pmatrix}a_{1}\operatorname{Id}_{k}&&-X\\ &a_{1}\operatorname{Id}_{k}&a_{2}\operatorname{Id}_{k}\\ a_{2}\operatorname{Id}_{k}&X&0\end{pmatrix}

gives a $k^{2}+2$ dimensional space of $3k\times 3k$ matrices of bounded rank $2k$ . L. Manivel points out that this suggests (after permuting the blocks):

\begin{pmatrix}a_{1}\operatorname{Id}_{k}&a_{2}\operatorname{Id}_{k}&a_{3}\operatorname{Id}_{k}\\ X&&&(a_{2}-a_{3})\operatorname{Id}_{k}\\ &X&&(a_{3}-a_{1})\operatorname{Id}_{k}\\ &&X&(a_{1}-a_{2})\operatorname{Id}_{k}\end{pmatrix}

and more generally $p$ blocks with $X$ and $(a_{1},\dotsc,a_{k})$ , $(b_{1},\dotsc,b_{k})$ linear forms such that

(a_{1},\dotsc,a_{k})\begin{pmatrix}b_{1}\\ \vdots\\ b_{k}\end{pmatrix}=0.

There is a unique up to isomorphism concise tensor in $\mathbb{C}^{3}{\mathord{\otimes}}\mathbb{C}^{3}{\mathord{\otimes}}\mathbb{C}^{3}$ that is both of minimal border rank and gives rise to a space of bounded rank [7], the space is

\begin{pmatrix}a_{1}&a_{2}&a_{3}\\ &a_{1}&\\ &a_{3}&\end{pmatrix}

(the tensor is $1_{B}$ and $1_{C}$ -generic, and $1_{A}$ -degenerate). Applying the construction of Proposition 2.3 with the other tensor $T_{skewcw,2}$ , yields a bounded rank $6$ space in $\mathbb{C}^{9}{\mathord{\otimes}}\mathbb{C}^{9}$ . This space is compression, but it may be useful for Strassen’s laser method. It also shows that even if the space of $k\times k$ matrices is of bounded rank less than $k$ , one can still obtain a bounded rank $kr$ space with the construction.

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On Linear spaces of of matrices bounded rank

Abstract.

2010 Mathematics Subject Classification:

1. Introduction

Example 1.1.

Example 1.2.

Example 1.3.

Acknowledgements

2. Results

Theorem 2.1.

Question 2.2.

Proposition 2.3.

Remark 2.4.

Remark 2.5.

Overview

3. Interpretations and generalizations via the associated tensors

3.1. Background

3.2. Motivations for this project

Strassen’s laser method

Geometric rank and cost v. value in quantum information theory

3.3. Brief discussion of tensors and their geometry

3.4. A geometric interpretation of the tensor associated to Case (IV) and a tensor variant of Proposition 2.3

Proposition 3.1.

Proof.

3.5. A geometric interpretation of Case (III) and generalizations

Proposition 3.2.

Proof.

3.6. Degeneracy loci and symmetry Lie algebras

Question 3.3.

Question 3.4.

Remark 3.5.

3.7. Border ranks

Proposition 3.6.

Proof.

Remark 3.7.

Proposition 3.8.

Proof.

4. Review of previous work

4.1. Work of Atkinson-Llyod and Atkinson [3, 2]

4.2. Work of Eisenbud and Harris [12]

Remark 4.1.

Proposition 4.2.

Proposition 4.3.

5. Towards uniting the linear algebra and algebraic geometry perspectives

5.1. Atkinson normal form

Lemma 5.1.

Proof.

Proposition 5.2.

Proof.

Proof of Proposition 2.3.

Definition 5.3.

Proposition 5.4.

Proof.

Corollary 5.5.

Proof.

Proposition 5.6.

Proof.

5.2. Chern classes and Atkinson numbers

Proposition 5.7.

Proof of Proposition 5.7.

Example 5.8.

5.3. Chern class defects

Proposition 5.9.

Proof of Proposition 5.9.

Remark 5.10.

Example 5.11.

6. Proof of the Main Theorem

6.1. Zero sets and syzygies of spaces of quadrics

6.2. Conclusion of the (5,5)(5,5) case

Lemma 6.1.

Proof.

6.3. (r+1)×m(r+1)\times m spaces of bounded rank rr

Theorem 6.2.

Proposition 6.3.

6.4. 2×k2\times k spaces of linear forms and their linear annihilators

6.5. (6,6)(6,6)-spaces

6.6. (6,5)(6,5) case

6.7. Good Chern classes

6.8. Proof that our two new examples are basic

Proof.

6.2. Conclusion of the $(5,5)$ case

6.3. $(r+1)\times m$ spaces of bounded rank $r$

6.4. $2\times k$ spaces of linear forms and their linear annihilators

6.5. $(6,6)$ -spaces

6.6. $(6,5)$ case