# The symmetric functions catalog

An overview of symmetric functions and related topics

2022-10-07

## Gelfand–Tsetlin patterns

Gelfand–Tsetlin (sometimes spelled Cetlin or Zetlin) patterns were introduced by I. Gelfand and M. Tsetlin in [GT50], motivated by the close relationship with representation theory of $GL_n.$ A short introduction to GT-patterns can be found in [p.103, Sta01].

A Gelfand-Tsetlin pattern or GT-pattern is a triangular (or sometimes parallelogram) arrangement of non-negative integers, with certain constraints. The triangular patterns are indexed as follows.

$\begin{matrix} x_{n1} & & x_{n2} & & \cdots & & \cdots & & x_{nn} \\ & \ddots & & \ddots & & & \iddots & & & \\ % & & x_{13} & & x_{23} & & x_{33}\\ % & & & x_{12} & & x_{22} \\ & & x_{21} & & x_{22}\\ & & & x_{11} \end{matrix}$

The less common parallelogram patterns are indexed as follows.

$\begin{matrix} x_{n1} & & x_{n2} & & \cdots & & \cdots & & x_{nm} \\ & \ddots & & \ddots & & & & & & \ddots \\ % & & x_{13} & & x_{23} & & x_{33} & & \\ % & & & x_{12} & & x_{22} & & & \\ & & x_{11} & & x_{12} & & \cdots & & \cdots & & x_{1m} \\ & & & x_{01} & & x_{02} & & \cdots & & \cdots & & x_{0m} \end{matrix}$

The entries must satisfy the conditions

\begin{equation*} x_{i+1,j} \geq x_{ij} \text{ and } x_{ij} \geq x_{i+1,j+1} \label{eq:gtinequalities} \end{equation*}

for all values of $i,$ $j$ where the indexing is defined. The inequalities simply states that horizontal rows are weakly decreasing, down-right diagonals are weakly decreasing and down-left diagonals are weakly increasing.

The weight $w(G) = (w_1,\dotsc,w_n)$ of a GT-pattern $G$ is defined as the vector where $w_i$ is the difference of the sum of row $i$ and $i-1.$ By convention, $x_{0j}=0$ if these entries are not specified.

### Bijection with semi-standard Young tableaux

The conditions ensures every row in the pattern defines an integer partition. Furthermore, any two adjacent rows define a skew shape. These properties enables us to biject triangular GT-patterns with Young tableaux and parallelogram GT-patterns with skew Young tableaux.

The skew shape defined by row $j$ and $j+1$ in a GT-pattern $G$ describes which boxes in a tableau $T$ that have content $j.$ In particular, if $T$ has shape $\lambda/\mu$ then the first row in $G$ is $\mu$ and the last row is $\lambda.$

Example.

The following triangular GT-pattern

$\begin{matrix} 5 & & 4 & & 2 & & 1 & & 1 & & 0\\ & 5 & & 3 & & 2 & & 1 & & 0\\ & & \color{blue} 3 & & \color{blue} 3 & & \color{blue} 2 & & \color{blue} 1 \\ & & & 3 & & 3 & & 1 & \\ & & & & 3 & & 2 & \\ & & & & & 3 & \\ \end{matrix}$

is in bijection with the semi-standard Young tableau below.

Notice that the fourth row in the GT-pattern is $3321$ and the shape of all entries less than or equal to four in the tableau is also $3321.$

Example.

The skew Young tableau

is in bijection with the GT-pattern

$\begin{matrix} 4 & & 3 & & 2 & & 1\\ & 4 & & 3 & & 1 & & 1\\ & & 3 & & 3 & & 1 & & 1\\ & & & 3 & & 2 & & 1 & & 0\\ & & & & 2 & & 1 & & 1 & & 0\\ \end{matrix}$

Observe that in any GT-pattern, $x_{i+1,j} - x_{i,j}$ counts the number of boxes with content $i$ in row $j$ in the corresponding tableau. Thus, the weight of a GT-pattern is the same as the weight of the corresponding tableau.

There is a crystal graph structure on semi-standard Young tableaux. The corresponding crystal graph on GT-patterns is described explicitly in https://arxiv.org/pdf/2005.06639.pdf.

### Tokuyama's formula

Let $SGT(\lambda)\subset \gtp(\lambda)$ be the set of strict Gelfand–Tsetlin patterns. This is the subset of patterns where each row is strictly decreasing. Consider tuples $(a,b,c)$ in the pattern arranged as $\begin{smallmatrix} a && b\\ & c \end{smallmatrix}.$ Let $S(G)$ be the number of entries $c$ such that $a\gt c \gt b,$ and let $L(G)$ be the number of entries $c$ such that $a=c.$

Let $\rho = (n-1,n-2,\dotsc,1,0).$ Then (see [Tok88])

\begin{equation*} \schurS_{\lambda}(x_1,\dotsc,x_n) \cdot \prod_{1 \leq i \lt j \leq n} (x_i + t x_j) = \sum_{G \in SGT(\lambda + \rho)} (1+t)^{S(G)} t^{L(G)}\xvec^{w(G)}. \end{equation*}

Note that this interpolates between the Weyl character formula, and the SSYT definition of Schur polynomials.

See also [Sta86a] for some special cases, and connection with plane partitions and alternating sign matrices.

## Gelfand–Tsetlin polytopes

By specifying the top row in a triangular GT-pattern as $\lambda$ and imposing the inequalities above we get a convex polytope. This is a Geltand–Tsetlin polytope, $\gtp_{\lambda} \subset \setR^{n(n+1)/2}.$ By construction, the integer lattice points in $\gtp_{\lambda}$ is in bijection with $\SSYT(\lambda).$ The analogous statement holds for parallelogram GT-patterns where also the bottom row $\mu$ is a fixed integer partition. These polytopes are denoted $\gtp_{\lambda\mu} \subset \setR^{nm}.$ Below, we use $d$ to denote the dimension ($n(n+1)/2$ or $mn$) of the ambient space.

### Unimodular triangulation and marked posets

The polytopes $\gtp_{\lambda\mu}$ admit unimodular triangulations. This means that the polytopes can be decomposed into simplices with integer vertices and normalized volume $1.$ For a short proof, see [Ale19b].

As a consequence, Gelfand–Tsetlin polytopes have integer vertices, and have the integer decomposition property (IDP). This means that if $p$ is a lattice point in $k \gtp_{\lambda\mu}$ for some integer $k\geq 1,$ then $p$ can be expressed as $p_1 + \dotsb + p_k$ where each $p_i$ is a lattice point in $\gtp_{\lambda\mu}.$ This can be proved directly by bijecting to semi-standard tableaux, see [Subsection 2.3, Ale16a].

A. Tsuchiya generalize the notion of IDP to $k$-tuples of polytopes (the $k=2$ case was introduced in [HH17]). The following theorem follows very easy from the bijection with SSYT.

Let $\gtp_{\lambda^1},\dotsc,\gtp_{\lambda^k}$ be GT-polytopes, embedded in the same ambient space by padding the partitions with zeros of needed. Then

$\setZ^d \cap \left( \gtp_{\lambda^1} + \dotsb + \gtp_{\lambda^k} \right) = \left( \setZ^d \cap \gtp_{\lambda^1} \right) + \dotsb + \left(\setZ^d \cap \gtp_{\lambda^k} \right).$

By letting all $\lambda^i$ be equal, we recover the usual IDP.

This theorem does not generalize to skew shapes. The pair $\gtp_{3}$ and $\gtp_{2/1}$ yields a counter-example.

Gelfand–Tsetlin polytopes are special cases of so called marked order polytopes, which generalizes order polytopes considered by R. Stanley, [Sta86b]. In [ABS11], the authors generalize the bijection between order polytopes and chain polytopes and give a marked chain polytope analogue of GT-polytopes. In type $A$ these analogues are so called Feigin–Fourier–Littelmann polytopes.

The Gelfand–Tsetlin polytopes are also special cases of so called flow polytopes, see [LMD19a].

### Ehrhart polynomials

By definition, it follows that for a partition of length $m,$

$\left| \setZ^d \cap \gtp_{\lambda} \right| = \schurS_{\lambda}(1^m) = \prod_{1 \leq i \lt j \leq m } \frac{\lambda_i - \lambda_j + j-i}{j-i},$

where the last identity is due to Stanley's hook-content formula. Furthermore, the Ehrhart polynomial of $\gtp_{\lambda}$ is given by

$Ehr(\gtp_{\lambda};k) = \prod_{1 \leq i \lt j \leq m } \frac{k(\lambda_i - \lambda_j) + j-i}{j-i}.$
Proposition.

We have that $1 + \sum_{n\geq 1} t^n |\schurS_{\lambda/\mu}(1^m)| = \frac{ \sum_{T \in \SYT(\lambda/\mu)} z^{\des(T)} }{(1-t)^{1+|\lambda/\mu|}}.$

In particular, the $h^*$-polynomial of $\gtp_{a^b}$ is the descent-generating polynomial of rectangular standard Young tableaux; $\sum_{T \in \SYT(a^b)} z^{\des(T)}.$

Proof.

Use [Eqs. (7.96), (7.108), Sta01], where one needs to note that GT-patterns for rectangular SSYTs can be made in bijection with rectangular plane partitions.

The coefficients of the Ehrhart polynomial of $\gtp_{\lambda\mu}$ has non-negative coefficients. Equivalently, the function

$f(k) = \schurS_{k\lambda/k\mu}(1^m)$

is a polynomial in $k$ with non-negative coefficients.

For example, $m=3,$ $\lambda=321$ and $\mu=1$ gives the Ehrhart polynomial

$Ehr(\gtp_{321/1},k) = \frac{1}{2} (k^5+6 k^4+14 k^3+16 k^2+9 k+2).$

### Vertices, $f$-vectors and differential equations

Ther has been some effort in counting the number of vertices of GT-polytopes. In [GKT13], the authors describe a generating function that enumerates the vertices of $\gtp_{\lambda}.$

Let $V(1^{i_1},2^{i_2},\dotsc,k^{i_k})$ denote the number of vertices of $\gtp_{\lambda}$ where $\lambda$ is the partition with type $(i_1,i_2,\dotsc,i_k).$ Consider the generating function

$E_k(z_1,\dotsc,z_k) \coloneqq \sum_{i_1,\dotsc,i_k \geq 0} V(1^{i_1},2^{i_2},\dotsc,k^{i_k}) \frac{z_1^{i_1}}{i_1!} \dotsm \frac{z_k^{i_k}}{i_k!}.$
Theorem (See [GKT13]).

The formal power series $E_k(z_1,\dotsc,z_k)$ satisfies the partial differential equation

$\left( \frac{\partial^k}{\partial z_1 \dotsm \partial z_k} - \left(\frac{\partial}{\partial z_1}-\frac{\partial}{\partial z_2}\right) \dotsm \left(\frac{\partial}{\partial z_{k-1}}-\frac{\partial}{\partial z_k}\right) \right)E_k = 0$

This result is later improved in [Thm. 1.14, ACK18], where the authors provide a differential equation for the full $f$-vector of GT-polytopes. They consider the generating function

$F_k(z_1,\dotsc,z_k;t) \coloneqq \sum_{d\geq 0} \sum_{i_1,\dotsc,i_k \geq 0} t^d f_d(1^{i_1},2^{i_2},\dotsc,k^{i_k}) \frac{z_1^{i_1}}{i_1!} \dotsm \frac{z_k^{i_k}}{i_k!}.$

where $f_d$ is the number of $d$-dimensional faces of $\gtp_{\lambda}.$

### Type B and C

There are analogues of GT-polytopes for type $B$ and $C,$ see [ABS11].

## Weight-restricted Gelfand–Tsetlin polytopes

Let $\gtp_{\lambda,w}\subseteq \gtp_{\lambda}$ be the Gelfand–Tsetlin polytope defined as $\gtp_{\lambda}$ but with the additional constraint that the difference of the row row sums of row $j+1$ and $j$ is exactly $w_j.$

By construction, lattice points in $\gtp_{\lambda,w}$ are in bijection with $\SSYT(\lambda,w).$ Therefore, the number of lattice points in $\gtp_{\lambda,w}$ is given by the Kostka coefficient $K_{\lambda w}.$

We do the analogous definitions for parallelogram GT-patterns and skew shapes.

### Non-integrality and vertices

Contrary to $\gtp_{\lambda},$ the polytopes (also called Gelfand–Tsetlin polytopes) $\gtp_{\lambda,w}$ are not always integral. In fact, the denominators of the vertices can get arbitrary large, see [DLM04a].

In [Ale16a], we prove that for $\lambda \vdash n,$ the polytope $\gtp_{\lambda,1^n}$ is non-integral unless $\lambda=(2,2)$ or a hook shape.

### Saturation theorem

The following theorem is a GT-polytope analogue of the celebrated Saturation theorem, proved by A. Knutson and T. Tau, . They proved a similar statement for so called Berenstein–Zelevinsky polytopes.

Theorem (See [Ale15]).

Every non-empty Gelfand–Tsetlin polytope $\gtp_{\lambda/\mu,w}$ contains at least one vertex with integer coordinates.

### Ehrhart polynomials and period collapse

Since the $\gtp_{\lambda/\mu,w}$ are not lattice polytopes, one does not expect the Ehrhart function to be a polynomial. However, these polytopes exhibit a period collapse [HM08] and they do have a polynomial Ehrhart function. See [Ras04a] for a proof of this statement.

Conjecture (King, Tollu, Toumazet 2004, [KTT04]).

The coefficients of the Ehrhart polynomial of $\gtp_{\lambda/\mu,w}$ are non-negative.

## Monotone triangles and alternating sign matrices

Monotone triangles are special types of triangular GT-patterns with additional constraints. First, the top row is $(n,n-1,\dotsc,1)$ and all rows must be strictly decreasing.

Monotone triangles of size $n$ are in bijection with $n \times n$ alternating sign matrices. Alternating sign matrices are matrices with entries in $\{-1,0,1\}$ such that in each row and column, the nonzero elements alternate in sign. Furthermore, the sum of the entries in each row (and column) is $1.$

Example.

Consider the following $5\times 5$ alternating sign matrix, where we omit the zeros, and $-1$ is written as $\color{blue}{\overline{1}}.$ The rows are accumulated from the bottom, so that row $k$ from the bottom is the sum of the first $k$ rows.

$\begin{bmatrix} &1& & & \\ 1&\color{blue}{\overline{1}}& &1& \\ &1& &\color{blue}{\overline{1}}&1\\ & &1& & \\ & & &1& \end{bmatrix} \to \begin{bmatrix} 1&1&1&1&1 \\ 1& &1&1&1 \\ &1&1& &1 \\ & &1&1& \\ & & &1& \end{bmatrix}$

For each row, we then record the columns that contains a $1$ and write the column indices in a decreasing fashion. The result is a monotone triangle.

$\begin{matrix} 5&&4&&3&&2&&1 \\ &5&&4&&3&&1 \\ &&5&&3&&2 \\ &&&4&&3 \\ &&&&4& \\ \end{matrix}$

D. Zeilberger [Zei95] proved that the number of monotone triangles of size $n,$ or equivalently, number of $n\times n$ alternating sign matrices is given by

$\prod_{j=0}^{n-1}\frac{(3j+1)!}{(n+j)!}.$

A generalization of this result is given in [Fis06], where difference operators are used in a formula to count the number of monotone triangles with prescribed top row $\lambda.$

## Applications

### Schur polynomials

The close connection between GT-patterns and Schur polynomials is quite apparent. The Schur polynomial $\schurS_\lambda(x_1,\dotsc,x_n)$ can be expressed as a sum over $GT(\lambda),$ the set of triangular GT-patterns with top row equal to $\lambda.$ We have that

$\schurS_\lambda(x_1,\dotsc,x_n) = \sum_{G \in GT(\lambda) } \xvec^{w(G)}$

where $w$ depends on the row-sums of the pattern $G.$

### Hall–Littlewood polynomials

The identity for Schur functions can be generalized to Hall–Littlewood polynomials, via a weighted version of Brion's theorem.

Theorem (See [FM16]).

Let $\lambda$ be a partition. Then

\begin{equation*} \hallLittlewoodP_\lambda(\xvec;t) = \sum_{G \in GT(\lambda) } p_G(t) \xvec^{w(G)} \end{equation*}

where $p_G(t)$ is a certain statistic that depends on $G.$

This theorem can also be proved via other means, see [Mac95].

### Key polynomials

The key polynomials generalize the Schur polynomials. In [KST12], a formula for key polynomials is proved, where the sum runs over lattice points in a union of certain faces of $GT(\lambda),$ see key polynomials from GT-patterns for details.

### Schubert polynomials

It was recently proved in [LMD19b] that Schubert polynomials can be expressed as a sum over lattice points in a Minkowski-sum of GT-polytopes.