Explicit constructions of connections on the projective line with a maximally ramified irregular singularity

The authors received support from the SQuaRE program of the American Institute of Mathematics and are grateful to AIM for its hospitality. The authors would like to thank the American Institute of Mathematics for its hospitality during an AIM SQuaRE where much of the work for this paper was completed. N.L. received support from the Roux Institute and the Harold Alfond Foundation. D.S.S. received support from Simons Collaboration Grant 637367. B.N. received support from an AMS–Simons Travel Grant.

We define a partition of a positive integer $n$ to be a multiset of positive integers that sum to $n$. Let $\operatorname{\mathrm{m}}_{\lambda}(x)$ denote the multiplicity in a partition $\lambda$ of an integer $x$. We allow the multiplicity to take the value zero. Define the support $\operatorname{\mathrm{Supp}}(\lambda)$ of a partition $\lambda$ to be the set of integers with positive multiplicity in $\lambda$; i.e., $\operatorname{\mathrm{Supp}}(\lambda)=\{x\in\mathbb{Z}_{>0}:\operatorname{\mathrm{m}}_{\lambda}(x)>0\}$. If $\operatorname{\mathrm{Supp}}(\lambda)=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{k}\}$, then we represent the partition $\lambda$ as $\{\lambda_{1}^{\operatorname{\mathrm{m}}_{\lambda}(\lambda_{1})},\lambda_{2}^{\operatorname{\mathrm{m}}_{\lambda}(\lambda_{2})},\ldots,\lambda_{k}^{\operatorname{\mathrm{m}}_{\lambda}(\lambda_{k})}\}$. Define $|\lambda|$ to be the sum $\sum_{x\in\operatorname{\mathrm{Supp}}(\lambda)}\operatorname{\mathrm{m}}_{\lambda}(x)$. Thus, if $\lambda$ is a partition of $n$, then $\sum_{x\in\operatorname{\mathrm{Supp}}(\lambda)}\left(\sum_{i=1}^{\operatorname{\mathrm{m}}_{\lambda}(x)}x\right)=n$; each of the $|\lambda|$ summands in the expansion of this double summation is called a part in $\lambda$.

We find it convenient to sometimes view a partition $\lambda$ as a monotonically decreasing tuple $(\lambda_{i})_{i=1}^{|\lambda|}$ of the parts in $\lambda$. The dominance order (also known as the majorization order, e.g., [22]) on the set of partitions of $n$ is the partial order $\succeq$ defined by $\lambda\succeq\mu$ if and only if $|\lambda|\leq|\mu|$ and $\sum_{i=1}^{j}\lambda_{i}\geq\sum_{i=1}^{j}\mu_{i}$ for all $j\in[1,|\lambda|]$.

If $R$ is a commutative ring with unity, then the general linear group $\operatorname{GL}_{n}(R)$ of degree $n$ over $R$ is the group of $n\times n$ matrices over $R$ with invertible determinant. Its Lie algebra $\operatorname{\mathfrak{gl}}_{n}(R)=\operatorname{\mathrm{Lie}}(\operatorname{GL}_{n}(R))$ is the vector space of $n\times n$ matrices over $R$, equipped with the Lie bracket $[\cdot,\cdot]$ defined by $[X,Y]=XY-YX$. We also view elements of $\operatorname{\mathfrak{gl}}_{n}(R)$ as endomorphisms of $R^{n}$. The adjoint action of $\operatorname{GL}_{n}(R)$ on $\operatorname{\mathfrak{gl}}_{n}(R)$ is defined by $\mathrm{Ad}_{g}(X)=gXg^{-1}$, for any $g\in\operatorname{GL}_{n}(R)$ and $X\in\operatorname{\mathfrak{gl}}_{n}(R)$. In this paper, the term “adjoint orbit” always refers to a complex adjoint orbit. We denote the adjoint orbit of an element $X\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ by $\mathscr{O}_{X}$; i.e., $\mathscr{O}_{X}=\{\mathrm{Ad}_{g}(X)\mid g\in\operatorname{GL}_{n}(\mathbb{C})\}$.

An element $X\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ is nilpotent if $X^{m}=0$ for some $m>0$. The adjoint orbit $\mathscr{O}_{X}$ of a nilpotent element $X\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ is called a nilpotent orbit. Any nilpotent orbit $\mathscr{O}$ contains a block-diagonal representative in Jordan canonical form, which is unique up to permutations of its blocks (known as “Jordan blocks”). Hence, there is a bijective correspondence between the set of nilpotent orbits in $\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ and the set of partitions of $n$, where each nilpotent orbit corresponds with the multiset of its Jordan block sizes. We say that $\mathscr{O}$—and each element of $\mathscr{O}$—has type $\lambda$ if the sizes of the Jordan blocks for $\mathscr{O}$ are given by the partition $\lambda$.

The set of nilpotent orbits are partially ordered: $\mathscr{O}^{1}$ is less than or equal to $\mathscr{O}^{2}$ if and only if $\mathscr{O}^{1}$ is contained in the Zariski closure of $\mathscr{O}^{2}$. This partial order is known as the closure order. The bijection between nilpotent orbits and partitions described above defines a poset isomorphism when the sets are endowed with the closure order and the dominance order respectively [8].

Next, we define two nilpotent matrices that arise in the study of Coxeter connections (see Section 3).

The following lemma shows that the orbit $\mathscr{O}_{N_{r}}$ of $N_{r}$ is the minimal orbit with $r$ blocks and its closure consists of all orbits with at most $r$ blocks. The proof is given in [24, §2].

Let $V$ be a rank $n$ trivializable vector bundle over the Riemann sphere $\mathbb{P}^{1}$ endowed with a meromorphic connection $\nabla$. After fixing a trivialization, the connection has the form $d+M(z)\frac{dz}{z}$, where $M(z)\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C}(z))$ is an $n\times n$ matrix of rational functions. The local behavior at $y\in\mathbb{P}^{1}$ is determined by the associated formal connection at $y$. Explicitly, this is obtained by expanding $M(z)$ as a Laurent series in term of a local parameter $u$ at $y$: $z-y$ if $y$ is finite and $z^{-1}$ otherwise (i.e., if $y=\infty$). Changing the trivialization of the global or formal vector bundle induces an action on the connection matrix called gauge change. If $d+A(u)\frac{dz}{z}$ is a formal connection with $A(u)\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C}(\!(u)\!))$, then $g\in\operatorname{GL}_{n}(\mathbb{C}(\!(u)\!))$ acts on the connection operator via $g\cdot(d+A(u))\frac{du}{u}=d+(gA(u)g^{-1})\frac{du}{u}-(dg)g^{-1}$. In the global case, $g\in\operatorname{GL}_{n}(\mathbb{C})$, so the nonlinear gauge term is zero.

Let $y$ be a singular point of the connection, and denote the induced formal connection at $y$ by

where $M_{i}\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ and $r\geq 0$. When the leading term $M_{-r}$ is well-behaved, it gives important information about the connection. For example, if $M_{-r}$ is not nilpotent, then the integer $r$ is an invariant of the connection known as the slope at $y$. The slope can roughly be viewed as a measure of the irregularity of the singularity; a singular point $y$ is regular singular if the slope is zero, and irregular otherwise. If $M_{-r}$ is regular semisimple (i.e., diagonalizable with distinct eigenvalues), then a classical result due to Wasow [29] states that the connection is locally gauge equivalent to a connection of the form

where $D_{i}\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ are diagonal and $D_{-r}$ is regular semisimple. Diagonal representatives of this form are called regular unramified formal types.²²2Any formal connection can be put into upper triangular form after passing to a finite extension of $\mathbb{C}(\!(u)\!)$. It is called unramified if this can be done over the ground field and ramified otherwise.

However, there are many singularities for which the leading term of this expansion is nilpotent, regardless of the choice of formal trivialization. Indeed, the slope at $y$ can be any nonnegative rational number with denominator at most $n$, and if this slope is not an integer, the leading term will always be nilpotent. For example, the Frenkel–Gross connection [10]

has two singular points, a regular singular point at $\infty$ and an irregular singular point at $0$ with slope $1/n$. This is the smallest possible slope of an irregular singularity [17]. The Frenkel–Gross connection is maximally ramified at $y=0$, meaning that the slope has the largest possible denominator (i.e., $n$) when reduced to lowest terms.

To better understand formal connections with nonintegral slope, Bremer and Sage developed a generalization of leading terms known as fundamental strata [6, 4, 7, 5]. Every formal connection “contains” a fundamental stratum [6, Lemma 4.5], and the slope of a connection is equal to the “depth” of any fundamental stratum it contains ([6, Theorem 4.10],[7, Theorem 2.14]). Regular semisimple leading terms are generalized by regular strata, and a generalization of Wasow’s result states that any connection containing a regular stratum can be “diagonalized” into a ramified formal type [6, 5]. In particular, if a singularity has slope equal to $r/n$ for some $r$ relatively prime with $n$—i.e., if the singularity is maximally ramified—then the connection is locally gauge equivalent to a connection of the form $d+p(\omega^{-1})\frac{dz}{z}$, where $p$ is a polynomial of degree $r$ and $\omega^{-1}=E_{1}z^{-1}+N_{1}$ [24, Theorem 4.4]. The one-forms $p(\omega^{-1})\frac{dz}{z}$ are the “maximally ramified formal types of slope $r/n$”.

An important problem in the study of meromorphic connections is the extent to which a globally defined connection is determined by its local behavior. Here, “local behavior” consists of:

• •

  a nonempty, finite set of irregular singular points $\{x_{i}\}_{i}$;

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  a corresponding collection $\mathbf{A}=(\mathscr{A}_{i})_{i}$ of formal types³³3A formal type may be viewed as a rational canonical form for a formal connection. See [27] for more details. $\mathscr{A}_{i}$;

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  a finite set of regular singularities $\{y_{j}\}_{j}$ disjoint from $\{x_{i}\}_{i}$; and

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  a corresponding collection $\mathbf{O}=(\mathscr{O}_{j})_{j}$ of “nonresonant”⁴⁴4An adjoint orbit is called nonresonant if no two eigenvalues differ by a nonzero integer. adjoint orbits $\mathscr{O}_{j}$.

We consider the category $\operatorname{\mathcal{C}}(\mathbf{A},\mathbf{O})$ of connections $\nabla$ defined over the rank $n$, trivializable vector bundle $V$ on $\mathbb{P}^{1}$ satisfying the following properties:

• •

  $\nabla$ has an irregular singularity at each $x_{i}$, a regular singularity at each $y_{j}$, and no other singularities;

• •

  at each $x_{i}$, $\nabla$ is “framable” (see, e.g., [26]) and has formal type $A_{i}$; and

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  at each $y_{j}$, $\nabla$ has residue in $\mathscr{O}_{j}$.

Boalch gave a general construction of the corresponding moduli space $\operatorname{\mathcal{M}}(\mathbf{A},\mathbf{O})$ in the case that each of the formal types are diagonal with regular semisimple leading term [3, Proposition 2.1]. This construction was extended to include “toral” ramified formal types by Bremer and Sage [6, Theorem 5.6].

Here, we give a relatively simple version of this construction for the important special case of connections on $\mathbb{P}^{1}$ with a maximally ramified singularity at $0$, (possibly) a regular singularity at $\infty$, and no other singularities [24, Proposition 5.3]. Such connections are called Coxeter connections (for reasons discussed in [18, 24]); they include both the Frenkel–Gross and Airy connections.

Let $B\subseteq\operatorname{GL}_{n}(\mathbb{C})$ be the subgroup of upper triangular matrices. Its Lie algebra $\mathfrak{b}$ is the set of upper triangular matrices in $\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$. Let $I$ denote the “Iwahori subgroup” of $\operatorname{GL}_{n}(\mathbb{C}[\![z]\!])$ corresponding to $B$, and let $\mathfrak{i}$ denote its Lie algebra. Explicitly, this means that $I$ (resp. $\mathfrak{i}$) is the preimage of $B$ (resp. $\mathfrak{b}$) via the map $\operatorname{GL}_{n}(\mathbb{C}[\![z]\!])\rightarrow\operatorname{GL}_{n}(\mathbb{C})$ (resp. $\operatorname{\mathfrak{gl}}_{n}(\mathbb{C}[\![z]\!])\rightarrow\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$) induced by the “evaluation at zero” map $z\mapsto 0$.

Recall that the residue of a one-form $(\sum_{i=-r}^{\infty}M_{i}z^{i})\frac{dz}{z}$ with $r\geq 0$ is $M_{0}$.

Very little is known about these moduli spaces when ramified singular points are allowed. Indeed, it is not even known when these moduli spaces are nonempty; this is the Deligne–Simpson problem. The original Deligne–Simpson problem, involving connections with only regular singular points, was solved by Crawley–Boevey in 2003 [9]. The unramified version, where unramified singular points are allowed, was solved more recently by Hiroe [14]. Recently, we solved the Deligne–Simpson problem for Coxeter connections [24, Theorem 5.4] together with Kulkarni and Matherne. Theorem 3.2 is a restatement of this result for the special case of homogeneous Coxeter connections, i.e., Coxeter connections where the formal type $\mathscr{A}$ is a monomial. Without loss of generality, we can assume that $\mathscr{A}=\omega^{-r}\frac{dz}{z}$ (with $\gcd(r,n)=1$) [18].

While the results of [24] determine exactly when $\operatorname{\mathcal{M}}(\omega^{-r}\frac{dz}{z},\mathscr{O}_{\lambda})$ is nonempty, they do not provide an explicit element of the moduli space. When $r>n$, it is easy to find such an connection. Indeed, if $X\in\mathscr{O}_{\lambda}$ is strictly upper triangular (for example, if $X$ is the Jordan canonical form of $\mathscr{O}_{\lambda}$), then $d+(\omega^{-r}+X)\frac{dz}{z}$ has the desired local behavior. Constructing an explicit connection in the moduli space is more complicated for $r<n$ and $|\lambda|\leq r$, but it can be translated into a certain problem in matrix theory.

In linear algebra, an “upper matrix completion problem” is a problem where one studies various properties of the cosets $A+\mathfrak{u}$, where $A\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ and $\mathfrak{u}$ is the space of strictly upper triangular matrices. These problems have attracted great interest in the matrix theory community; see, for example, [1, 11, 12, 30, 2, 23, 25, 22, 21], as well as the survey in [16, Chapter 35]. Here, we are interested in the Upper Nilpotent Completion Problem: given $A$ nilpotent, determine the nilpotent orbits which intersect the coset $A+\mathfrak{u}$. The most general result on this problem was conjectured by Rodman and Shalom [25] and proved by Krupnik and Leibman [22]. It states that if $\mu$ is the partition of $n$ corresponding to the nilpotent orbit of $A$ and $\lambda$ is a partition dominating $\mu$, then there exists an element in $(A+\mathfrak{u})\cap\mathscr{O}_{\lambda}$.

Let us now restrict to the case $A=N_{r}$, and let $\mu_{r}$ be the corresponding partition. It is shown in [24] that the partitions dominating $\mu_{r}$ are precisely those with at most $r$ parts. Let $\lambda$ be such a partition. If $X$ is a strictly upper triangular matrix such that $N_{r}+X$ is nilpotent of type $\lambda$, then one can show that $d+(\omega^{-r}+X)\frac{dz}{z}$ corresponds to the element $B\cdot(\alpha,Y)\in\operatorname{\mathcal{M}}(\omega^{-r}\frac{dz}{z},\mathscr{O}_{\lambda})$ with $\alpha=(\omega^{-r}+X)\frac{dz}{z}$ and $Y=-(N_{r}+X)$. Krupnik and Leibman’s result guarantees the existence of such an $X$. However, their proof, while constructive in theory, does not seem possible to carry out in practice. Indeed, their algorithm requires repeatedly transforming matrices into special Jordan-like forms, and even when starting with $A=N_{r}$, we are not aware of a numerically stable way of carrying out this algorithm.

In the next section, we specify a numerically stable and highly efficient algorithm that generates explicit solutions to the Upper Nilpotent Completion Problem for $N_{r}$, and hence leads to the construction of explicit Coxeter connections with arbitrary specified local behavior.

Let $n$ be a positive integer, and let $V$ be a set of $n$ elements, called vertices. Fix a bijection $\mathrm{n}\colon V\rightarrow[1,n]$, called an ordinal function, which imposes a total ordering on $V$. We define a $\operatorname{\mathfrak{gl}}_{n}$-graph to be a pair consisting of:

1. (1)

   a subset $E\subseteq V\times V$, whose elements are called arrows; and

2. (2)

   a weight function $\alpha\colon E\rightarrow\mathbb{C}-\{0\}$.

We define a bijective correspondence $\mathrm{graph}$ between $\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ and the set of $\operatorname{\mathfrak{gl}}_{n}$-graphs as follows. Let $A=(\alpha_{i,j})_{i,j=1}^{n}$ be an element of $\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$. Define $\mathrm{graph}(A)$ to be the $\operatorname{\mathfrak{gl}}_{n}$-graph with the property that two vertices $u$ and $v$ are joined by an arrow $u\mapsto v$ if and only if $\alpha_{\mathrm{n}(v),\mathrm{n}(u)}\neq 0$, and the weight of each arrow $u\mapsto v$ is $\alpha_{\mathrm{n}(v),\mathrm{n}(u)}$. Define $\mathrm{matrix}$ to be the inverse of the function $\mathrm{graph}$.

We consider embeddings $\phi$ of $\operatorname{\mathfrak{gl}}_{n}$-graphs—or more precisely, their vertex sets—into $\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$. The image $\phi(v)=(x,y)$ of a vertex $v\in V$ has position $x=x(v)$ and level $y=y(v)$. Henceforth, we use the term “graph” to refer to a $\operatorname{\mathfrak{gl}}_{n}$-graph equipped with an embedding into $\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$. We frequently refer to a vertex in a graph $\Gamma$ via its coordinates, and write $\mathrm{n}_{\Gamma}(x,y)$ to denote the ordinal of $\phi^{-1}((x,y))$.

A vertex $v$ is above (resp. below) a vertex $w$ if $x(v)=x(w)$ and $y(v)=y(w)+1$ (resp. $y(v)=y(w)-1$). An arrow $v\mapsto w$ goes down if $y(v)>y(w)$, and goes down-right if $y(v)>y(w)$ and $x(v)\leq x(w)$. A graph $\Gamma$ is downward if each of its arrows go down. A graph $\Gamma$ is properly downward if:

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  $\Gamma$ is downward;

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  for every vertex $v$ with $y(v)>1$, there is a vertex $w$ below $v$ and an arrow $v\mapsto w$; and

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  every arrow $v\mapsto w$ with $y(v)=y(w)+1$ goes down-right.

We define the domain $\mathrm{Dom}(\Gamma)$ of a graph $\Gamma$ with vertex set $V$ by $\mathrm{Dom}(\Gamma)=\{x(v)\mid v\in V\}$. The column in a graph $\Gamma$ at position $i$—or, more concisely, column $i$ in $\Gamma$—is $\{v\in V\mid x(v)=i\}$. If $C$ is a column in a graph, then the height of $C$ is $\max\{y(v)\mid v\in C\}$. If $i\in\mathrm{Dom}(\Gamma)$, then we denote the height of column $i$ in $\Gamma$ by $\mathrm{ht}_{\Gamma}({i})$, and the multiset of heights in $\Gamma$ by $\mathrm{Part}(\Gamma)$.

We say that a column $C$ in a properly downward graph $\Gamma$ is a downward path in $\Gamma$ if the following condition holds: for each vertex $v\in C$, there exists an arrow $v\mapsto w$ if and only if $w$ is below $v$, and there exists an arrow $w\mapsto v$ if and only if $w$ is above $v$.

Let $X$ be a subset of the vertex set of a graph $\Gamma$. We say that $X$ is ordered by type-writer traversal if, for each pair $v,w$ of vertices in $X$, $\mathrm{n}(v)<\mathrm{n}(w)$ if and only if one of the following holds: $y(v)>y(w)$, or $y(v)=y(w)$ and $x(v)<x(w)$. Note that if any subset of vertices is removed from a set ordered by type-writer traversal, then the resulting set remains ordered by type-writer traversal.

Let $\Gamma$ be a graph, and let $\phi$ be the associated embedding of the vertex set into $\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$. Any change of the coordinates of the vertices of $\Gamma$ is called a geometric transformation of $\Gamma$. In other words, a geometric transformation of $\Gamma$ is the replacement of $\phi$ with another embedding $\phi^{\prime}$. Geometric transformations preserve the matrix associated to $\Gamma$. More generally, a transformation of a graph $\Gamma$ describes any finite sequence of the following:

• •

  an addition or deletion of an arrow $u\mapsto v$;

• •

  a change of the weight $\alpha_{\mathrm{n}(v),\mathrm{n}(u)}$ of an arrow $u\mapsto v$; or

• •

  a geometric transformation.

We focus on one class of graph transformations that we call “graft transformations”, defined in Definition 4.4. Our algorithm (Algorithm 1) performs a finite sequence of these graft transformations.

Our algorithm, Algorithm 1, is specified below. The algorithm takes as input two positive integers $r$ and $n$ with $r<n$, and a partition $\lambda$ of $n$ with at most $r$ parts. The algorithm consists of the following objects, each of which has a state that changes over the course of execution:

• •

  a graph $\Gamma$;

• •

  two integers $\mathit{s}$ and $\mathit{c}$ (which we call “column pointers”); and

• •

  two multisets $\mathsf{T}$ and $\mathsf{S}$.

The initial state of $\Gamma$ is the graph of $N_{r}\in\operatorname{\mathfrak{gl}}_{n}(\mathbb{C})$ as described in Lemma 4.3. The column pointers and multisets are initialized in lines 3–10. The algorithm makes use of three operations on multisets: $+$, $-$, and $\max$. To define these operations, let $\lambda$ and $\mu$ be multisets. The sum $\lambda+\mu$ is the multiset with multiplicity function $\operatorname{\mathrm{m}}_{\lambda+\mu}(x)=\operatorname{\mathrm{m}}_{\lambda}(x)+\operatorname{\mathrm{m}}_{\mu}(x)$ for all $x$. The difference $\lambda-\mu$ is the multiset with multiplicity function $\operatorname{\mathrm{m}}_{\lambda-\mu}(x)=\max\{0,\operatorname{\mathrm{m}}_{\lambda}(x)-\operatorname{\mathrm{m}}_{\mu}(x)\}$ for all $x$. Finally, $\max(\lambda)$ is the maximum value in $\operatorname{\mathrm{Supp}}(\lambda)$.

After initialization, Loop 1 (lines 11–32) and Loop 2 (lines 33–55) are executed. We refer to these two loops as the primary loops (as opposed to the “Little Loop” defined in lines 34–36). Each iteration of a primary loop performs at least one graft transformation to $\Gamma$, reduces the cardinality of $\mathsf{S}$ and/or $\mathsf{T}$ by one, and increases the column pointer $\mathit{c}$ (and possibly increases $\mathit{s}$ as well). In Section 5, we prove that Algorithm 1 terminates after some finite number of iterations of the primary loops, and that the terminal state of $\Gamma$ satisfies the specified postconditions.