Kohnert’s rule for flagged Schur modules

Consider the infinite, doubly indexed family of indeterminates $\{z_{i,j}\}_{i,j=1}^{\infty}$. A matrix $A\in\mathrm{GL}_{N}$ act on $\mathbb{C}[z_{i,j}]$ in the usual way,

A diagram $D$ is a finite collection of cells in $\mathbb{Z}_{+}\times\mathbb{Z}_{+}$. We use matrix convention, so that the northwest corner has index $(1,1)$. For example, Fig. 1 shows the Ferrers diagram for a partition, the skew diagram for nested partitions, the key diagram for a weak composition, the Rothe diagram for a permutation, and a generic diagram.

A tableau or filling on $D$ is any map $T:D\rightarrow\mathbb{N}$. Abusing notation, we let $D$ also denote the row tableau on the shape $D$ where each cell is assigned its row index. For example, Fig. 2 shows a generic tableau (left) and the row tableau (right).

For example, taking $T$ to be the left tableau in Fig. 2, we have

Notice $\Delta_{T}=0$ whenever $T$ has repeated column entries.

In particular, taking $D$ to be the Ferrers diagram of the partition $\lambda$ as in the leftmost diagram of Fig. 1, the Schur module $\mathcal{S}_{D}$ is the irreducible representation of $\mathrm{GL}_{n}$ indexed by $\lambda$, which we denote by $\mathcal{S}_{\lambda}$.

For $B\subset\mathrm{GL}_{n}$ the subalgebra of upper triangular matrices, there is a $B$-stable ideal $I=\langle z_{i,j}\mid i>j\rangle$. We consider the $B$-module spanned by those fillings that respect this ideal. Say a filling $T$ is flagged if $T_{i,j}\leq i$.

The character of a $B$-module is the trace of the action of the matrix $X=\mathrm{diag}(x_{1},\ldots,x_{n})$. Many flagged Schur modules arise naturally in the context of geometry: for $D$ a key diagram, the module $\mathcal{S}^{\mathrm{flag}}_{D}$ coincides with those introduced by Demazure [Dem74b, Dem74a] that give a filtration of irreducible modules with respect to the Weyl group; and for $D$ a Rothe diagram, the module $\mathcal{S}^{\mathrm{flag}}_{D}$ coincides with the Schubert modules introduced by Kraskiewicz and Pragacz [KP87, KP04] whose characters are Schubert polynomials of Lascoux and Schützenberger [LS82] that compute intersection multiplicities for the flag manifold.

In what follows, we study these modules through their characters, to wit the following results are useful.

The weight of a filling $T$ is the weak composition $\mathbf{wt}(T)$ whose $i$th part is the number of entries of $T$ with label $i$. Given $\mathbf{a}$, let $x^{\mathbf{a}}=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$.

Th following is used to determine when the character of a flagged Schur module differs from the Kohnert polynomial (defined below). Given position integers $r<s$, let $\alpha_{r,s}$ be the weak composition with $r$th entry $1$, $s$th entry $-1$, and all others $0$.

Each irreducible $\mathrm{GL}_{n}$-module $\mathcal{S}_{\lambda}$ decomposes into weight spaces as $\mathcal{S}_{\lambda}=\bigoplus_{\mathbf{a}}\mathcal{S}_{\lambda}^{\mathbf{a}}$. Demazure [Dem74a] considered the $B$-action on the extremal weight spaces of $S_{\lambda}$, those whose weak composition weight $\mathbf{a}$ is a rearrangement of the highest weight $\lambda$. The extremal weights are naturally indexed by the pair $(\lambda,w)$, where $\lambda$ is the partition weight and $w$ is the minimum length permutation that rearranges $\lambda$ to $\mathbf{a}$.

At the one extreme, the Demazure module $\mathcal{S}_{\lambda}^{\mathrm{id}}$ for the identity permutation is the one-dimensional subspace of $\mathcal{S}_{\lambda}$ containing the highest weight element. At the other extreme, the Demazure module $\mathcal{S}_{\lambda}^{w_{0}}$ for the long element of $\mathcal{S}_{n}$ is the full module $\mathcal{S}_{\lambda}$. In general, the Demazure modules give a filtration from the highest weight element, namely $\mathcal{S}_{\lambda}^{u}\subset\mathcal{S}_{\lambda}^{w}$ whenever $u\prec w$ in Bruhat order.

The characters of Demazure modules can be computed by degree-preserving divided difference operators [Dem74b], as well as myriad combinatorial models.

For a positive integer $i$, the divided difference operator $\partial_{i}$ is the linear operator that acts on polynomials $f\in\mathbb{Z}[x_{1},x_{2},\ldots]$ by

where $s_{i}\in\mathcal{S}_{\infty}$ is the simple transposition that exchanges $x_{i}$ and $x_{i+1}$. Fulton [Ful92] connected the divided difference operators with modern intersection theory, providing a direct geometric context for Schubert polynomials [LS82]. An isobaric variation, sometimes called the Demazure operator, is defined by

Given a permutation $w$, we may define

for any expression $s_{i_{k}}\cdots s_{i_{1}}=w$ with $k$ minimal. Such an expression is called a reduced expression for $w$. Both $\partial_{i}$ and $\pi_{i}$ satisfy the braid relations, and so are independent of the choice of reduced expression for $w$. In fact, the expression for $\pi_{w}$ need not be reduced, since $\pi_{i}(f)=f$ whenever $s_{i}\cdot f=f$.

In particular, when $s_{i}w\prec w$ in Bruhat order, we have the following identity,

For $\mathbb{D}(\mathbf{a})$ the key (left-justified) diagram of a composition $\mathbf{a}=w\cdot\lambda$, we have

In this sense, the flagged Schur modules generalize Demazure characters.

Both key diagrams indexing Demazure modules and Rothe diagrams indexing Kraskiewicz–Pragacz modules exhibit the following property.

Northwest in this context is equivalent to southwest as used by Assaf and Searles [AS19, Ass20b] for Kohnert polynomials.

Magyar [Mag98b] gave a recurrence for computing characters of flagged Schur modules for %-avoiding shapes (see Definition 4.13). To state the recurrence, let $C_{k}$ denote the diagram with cells in rows $1,2,\ldots,k$ of the first column. Given a diagram $D$, let $s_{r}D$ denote the diagram obtained from $D$ by permuting the cells in rows $r,r+1$. Restricting Magyar’s result to northwest shapes gives the following.

For example, Magyar’s recurrence allows us to compute the character for the flagged Schur module indexed by the leftmost diagram $T_{5}$ in Figure 3 as follows:

Two Demazure modules $\mathcal{S}_{\mu}^{u}$ and $\mathcal{S}_{\nu}^{v}$ correspond precisely when $u\cdot\mu=v\cdot\nu$. While this implies $\mu=\nu$, the permutations $u,v$ can differ. Therefore the more natural indexing set for Demazure characters is obtained by specifying the weak composition given by the permutation acting on the partition. Define

where $w$ sorts the weak composition $\mathbf{a}$ to the weakly decreasing partition $\lambda$.

Perhaps unsurprising when comparing (2.5) and (2.8), Reiner and Shimozono [RS98] use Magyar’s recurrence for the character of flagged Schur modules [Mag98b] to prove the characters of flagged Schur modules decompose as a nonnegative sum of Demazure characters; see [RS98] for precise definitions.

One consequence of our main result is a new proof of this positivity and a new combinatorial interpretation for the coefficients.

Kohnert gave a combinatorial model for Demazure characters [Koh91] in terms of the following operation on diagrams.

Given a diagram $D$, let $\mathrm{KD}(D)$ denote the set of diagrams that can be obtained by some sequence of Kohnert moves on $D$. For example, Fig. 4 shows the Kohnert diagrams for the bottom diagram, where an edge from $S$ up to $T$ indicates $T$ can be obtained from $S$ by a single Kohnert move.

Assaf and Searles define the Kohnert polynomial of any diagram as follows.

Assaf and Searles conjectured [AS19] and Assaf proved [Ass20b] that for $D$ northwest, the Kohnert polynomial expands nonnegatively into Demazure characters.

For the example, the Kohnert polynomial expands into Demazure characters as

Comparing Theorem 2.10 with Theorem 2.14, one begins to suspect the following.

We prove Conjecture 2.15 by showing Kohnert polynomials of northwest diagrams satisfy the recurrence in Theorem 2.9.

By definition, we have $\mathfrak{K}_{\varnothing}=1$, establishing Theorem 2.9(M1). It is straightforward to establish Theorem 2.9(M2).

We next consider $D$ and $s_{i}D$ when row $i$ is a subset of row $i+1$, meaning the set of occupied columns in row $i$ is a subset of the occupied columns of row $i+1$. These conditions combined imply all cells in row $i+1$ without cells above them in row $i$ lie to the right of all cells in row $i$. Otherwise, the cell in row $i+1$ without a cell above it in row $i$ and any cell to the right in row $i$ violate the northwest condition. This also implies $s_{i}D$ is northwest. The only violation must take place in rows $i$ and $i+1$, but we only move cells in row $i+1$ to row $i$ where all cells to the left in row $i+1$ have a cell in row $i$ above them.

Since $\pi_{i}$ acts on a polynomial by adding additional monomials, this makes Theorem 2.9(M3) plausible. However, in order to understand precisely how $\pi_{i}$ acts on Kohnert polynomials, we must rely on Demazure crystals.

A crystal graph [Kas90] is a combinatorial model for a highest weight module, consisting of a vertex set $\mathcal{B}$, corresponding to the crystal base (see also the canonical basis [Lus90]), and directed, colored edges, corresponding to deformations of the Chevalley generators. For a highest weight $\lambda$ for $\mathrm{GL}_{n}$, the crystal base for $\mathcal{S}_{\lambda}$ is naturally indexed by $\mathrm{SSYT}_{n}(\lambda)$, the semistandard Young tableaux of shape $\lambda$ with entries in $\{1,2,\ldots,n\}$. We adopt the French (coordinate) convention for $\mathrm{SSYT}$ in which entries weakly increase left to right within rows and strictly increase bottom to top within columns. Kashiwara and Nakashima [KN94] and Littelmann [Lit95] defined the crystal edges on tableaux as follows.

The raising and lowering operators on crystals are maps $e_{i},f_{i}:\mathcal{B}\rightarrow\mathcal{B}\cup\{0\}$.

We draw a crystal graph with an $i$-colored edge from $b$ to $b^{\prime}$ whenever $b^{\prime}=f_{i}(b)$, omitting edges to $0$. For examples, see Fig. 5.