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# 1. Definition

A

**Gelfand-Tsetlin pattern**(or GT-pattern) is a triangular configuration of non-negative integers,$$\begin{array}{cccccccccccccc} x_{n1} & & x_{n2} & & \cdots & & x_{nn} \\ & \ddots & & \ddots & & & & & \\ & & x_{21} & & x_{22} \\ & & & x_{11} & \end{array}$$

with $x_{i+1,j} \geq x_{ij}$ and $x_{ij} \geq x_{i+1,j+1}$ whenever the indexing is defined.

the 9 Gelfand-Tsetlin patterns of size (2, 4) | ||||||||

[[4,0],[0]] |
[[4,0],[1]] |
[[4,0],[2]] |
[[4,0],[3]] |
[[4,0],[4]] |
[[3,1],[1]] |
[[3,1],[2]] |
[[3,1],[3]] |
[[2,2],[2]] |

# 2. Additional information

The two defining conditions ensure every row to be an integer partition and that any two adjacent rows define a skew shape.

The

**shape**of a GT-pattern is the partition given by its top row and its**weight**(or**type**) of the integer partition of the multiplicities of its entries. The following GT-pattern has shape $(5,4,3,1)$ and weight $(4,4,3,1,1)$:

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

There is a natural bijection between GT-patterns with parameters $(l,s)$ and semi-standard Young tableaux with $s$ boxes and maximal entry at most $l$. The skew shape defined by row $j$ and $j+1$ (counting from the bottom) in a GT-pattern describes which cells in the corresponding tableau contain $j$.

# 3. Gelfand-Tsetlin polytopes

The inequalities above (with a fixed top row and fixed row sums) describe a polytope in $\mathbb{R}^{\frac{n(n+1)}{2}}$, and each GT-pattern therefore naturally corresponds to a lattice point inside this polytope. This type of polytope has integer vertices.

The usual definition of a *Gelfand-Tsetlin polytope* is to impose an additional restriction on the row sums. Thus, the GT-polytope $P_{\lambda,w}$ is defined as all real Gelfand-Tsetlin patterns $(x_{ij})$ such that $x_{ni}=\lambda_i$ and $\sum_i (x_{j,i}-x_{j-1,i}) = w_j$ where $x_{ij}=0$ whenever the indexing is outside the triangular array. Then, the lattice points in $P_{\lambda,w}$ are in bijection with semi-standard Young tableaux of shape $\lambda$ and type $w$, and the lattice points are counted by the Kostka numbers $K_{\lambda w}$.

The vertices of these polytopes are for small examples integral [KTT04], but there are families of GT-patterns where the GT-polytope has arbitrary large *denominator*, see [Lo04].

Hence, for each pair $(\lambda,w)$ the polytope can $P_{\lambda,w}$ either be

- Empty
- Integral
- Non-integral

All non-empty $P_{\lambda,w}$ contain at least one integral vertex, (this is a consequence of Knutson and Tao's proof of the Saturation conjecture). All polytopes $P_{\lambda,\lambda}$ consist of exactly one lattice point.

Since $K_{\lambda w}$ is invariant under permutation of the entries in $w$ (proved by Bender and Knuth), most research has been focused on the case when $w$ is a partition. However, $P_{\lambda,w}$ might be integral for some $w$ but non-integral for some permutation of w.

## 3.1. Tiles

In [Lo04], the notion of *tiles* is introduced as an important statistic on GT-patterns. The related statistic is linked to the dimension of faces of Gelfand-Tsetlin polytopes.

The *tiling* of a pattern is the finest partition of the entries in the pattern, such that adjacent (NW,NE,SW,SE) entries that are equal belong to the same part. These parts are called *tiles*, and each entry in a pattern belong to exactly one tile.

A tile is *free* if it do not intersect any of the first and the last row.

For example,

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

has 7 tiles, and the tile consisting of the two 3:s is the only free tile.

"The machinery of tilings allows us easily to find non-integral vertices of GT-polytopes by looking for a tiling with a tiling matrix satisfying certain properties. [See Lemma 2.2 in source] Then the tilings can be "filled" in a systematic way with the entries of a GT pattern that is a non-integral vertex." [Lo04]

# 4. Applications

## 4.1. Schur polynomials

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

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

where $w$ is the weight of the pattern $G$.

## 4.2. Hall–Littlewood polynomials

The identity for Schur functions can be generalized to Hall–Littlewood polynomials, via a weighted version of Brion's theorem. See [FM06]

Let $\lambda$ be a partition. Then

$$P_\lambda(x;t) = \sum_{G \in GT(\lambda) } p_G(t) x^{w(G)}$$

where $p_G(t)$ is a certain statistic that depends on $G$. This theorem can also be proved via other means, see [Mac95]

## 4.3. 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 so called *reduced Kogan faces* of $GT(\lambda)$.

## 4.4. Schubert polynomials

It was recently proved in 1903.08275 that *Schubert polynomials* can be expressed as a sum over lattice points in a Minkowski-sum of GT-polytopes.

Alternatively, Schubert polynomials can also be expressed as a sum over reduced Kogan faces of a certain Gelfand-Tsetlin polytope, where each face contribute with a single monomial. The reduced Kogan faces are in bijection with pipe dreams, which are used to describe Schubert polynomials.

# 5. Remarks

It is possible to extend this definition to a parallelogram GT-pattern, in which case the GT-pattern corresponds to a skew semi-standard Young tableau. For example, here is such a GT-pattern and the corresponding skew semi-standard Young tableau: $$\begin{matrix}5 & & 4 & & 3 & & 1\\ & 4 & & 4 & & 3 & & 1\\ & & 4 & & 4 & & 2 & & 1\\ & & & 4 & & 3 & & 2 & & 1\\ & & & & 3 & & 2 & & 2 & & 0\\ & & & & & 2 & & 2 & & 0 & & 0 \\ \end{matrix} \qquad \begin{matrix} & & 1 & 2 & 5\\ & & 2 & 3\\1 & 1 & 4\\2\\ \end{matrix}.$$

The skew shape is $(5, 4, 3, 1)/(2, 2)$ and the weight is $(3, 3, 1, 1, 1)$.

## 5.1. Alternating sign matrices

There is also a correspondence between Gelfand Tsetlin patterns and Alternating Sign Matrices. In this case we use a special type of Gelfand Tsetlin pattern called a *monotone triangle*. Monotone triangles are Gelfand Tsetlin patterns with strict inequalities instead of weak, and the top row equals $1 2 \cdots n$.

Using the alternating sign matrix and its monotone triangle as above we have a map which gives the following Gelfand Tsetlin pattern:

\begin{align*} 1 & &2 & &3 & &4 \\ &\; \; \; \; \; \; \;1 & &\; \; \; \; \; \; \;\;3 & &\; \; \; \; \; \; \;4 & \\ & & 1 & &3 & & \\ & & &\; \; \; \; \; \; \; \;2 & & & \end{align*} which has column partial sum matrix $\begin{pmatrix}0 &1 &0 &0\\1 &0 &1 &0 \\1 &0 &1 &1 \\1 &1 &1 &1 \end{pmatrix}$ and maps to the alternating sign matrix $\begin{pmatrix}0 &1 &0 &0 \\1 &-1 &1 &0 \\0 &0 &0 &1 \\0 &1 &0 &0 \end{pmatrix}$

# 6. Additional resources

See here an interesting video showing the bijection from a GT Pattern to a SSYT.

See [Sta99] p. 313 for further references.

Gelfand-Tsetlin patterns in PerAlexandersson's catalog of symmetric functions.

# 7. References

*A combinatorial formula for affine Hall–Littlewood functions via a weighted Brion theorem*, Selecta Mathematica, 22(3):1703–1747, March 2016.

[KST12] Valentina A. Kirichenko, Evgeny Yu Smirnov, and Vladlen A. Timorin, *Schubert calculus and Gelfand–Zetlin polytopes*, Russian Mathematical Surveys, 67(4):685, 2012.

[KTT04] R.C. King, C. Tollu, and F. Toumazet, *Stretched Littlewood-Richardson coefficients and Kostka coefficients*, CRM Proceedings and Lecture Notes (2004), no. 24, 99–112..

[Lo04] J. A. De Loera and T. B. McAllister, *Vertices of Gelfand-Tsetlin polytopes*, Discrete Comput. Geom. 32 (2004), no. 4, 459–470.

[Mac95] Ian G. Macdonald, *Symmetric functions and Hall polynomials*, The Clarendon Press Oxford University Press, New York, second edition, 1995.

[Sta99] R. Stanley, *Enumerative Combinatorics II*, Cambridge University Press (1999).

# 8. Sage examples

# 9. Technical information for database usage

A GT-pattern is uniquely represented as a

**list of lists**representing its rows from top to bottom.GT-patterns are parametrized by two parameters

**length of the first row**and**sum of the first row**.- The database contains all GT-pattern where the sum of the two parameters is at most 6.