Gelfand-Tsetlin Patterns

Contents

# 1. Definition

A **Gelfand-Tsetlin pattern** (or GT-pattern) is a triangular configuration of non-negative integers,

$$\begin{array}{cccccccccccccc} x_{n1} & & x_{n2} & & \cdots & & x_{nn} \\ & \ddots & & \ddots & & & & & \\ & & x_{21} & & x_{22} \\ & & & x_{11} &\end{array}$$

where the entries must satisfy the conditions $x_{i+1,j} \geq x_{ij}$ and $x_{ij} \geq x_{i+1,j+1}$ for all values of $i$, $j$ where the indexing is defined. The conditions ensure every row in the pattern defines an integer partition. Furthermore, any two adjacent rows define a skew shape.

There is a natural bijection between GT-patterns and semi-standard Young tableaux; The skew shape defined by row $j$ and $j+1$ in a GT-pattern describes which boxes in the corresponding tableau that have content $j$.

# 2. 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. Examples

The following GT-pattern has shape $(5,4,3,1)$ and weight (or type) $(4,4,3,1,1)$.

$$\begin{matrix}5 & & 4 & & 3 & & 1 & & 0\\ & 5 & & 4 & & 3 & & 0 \\ & & 5 & & 3 & & 3 & \\ & & & 5 & & 3 \\ & & & & 4\\ \end{matrix} \qquad \overset{bijection}{\longleftrightarrow} \qquad \begin{matrix}1 & 1 & 1 & 1 & 2\\2 & 2 & 2 & 4\\3 & 3 & 3\\5\\\end{matrix}.$$

See this video for an explanation of this bijection.

# 4. Properties

See [Sta99] p. 313 for further references.

# 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, $\begin{matrix}5 & & 4 & & 3 & & 1\\ & 4 & & 4 & & 3 & & 1\\ & & 4 & & 4 & & 2 & & 1\\ & & & 4 & & 3 & & 2 & & 1\\ & & & & 3 & & 2 & & 2 & & 0\\ & & & & & 2 & & 2 & & 0 & & 0\\\end{matrix}$ corresponds to the skew semi-standard Young tableau $\begin{matrix} & & 1 & 2 & 5\\ & & 2 & 3\\1 & 1 & 4\\2\\\end{matrix}$.

Here is a fun video showing the bijection from a GT Pattern to a SSYT: Video

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

There also exists to a correspondence from Gelfand Tsetlin Patterns to an 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. 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]

# 7. Statistics

We have the following **11 statistics** in the database:

# 8. Maps

We have the following **1 maps** in the database:

The following maps have not yet considered:

- There exists a map relating the first row of a Gelfand Tsetlin Pattern and the shape of a Semi-standard Tableaux.

This map can be found by looking at the top row (or first partition) of a GT Pattern. Moving from left to right, each element corresponds to the amount of numbers (per row) in a Semi-standard Tableaux.

- Given a Gelfand-Tsetlin Pattern with i entries in the first row, and i rows, there exists a map relating Gelfand Tsetlin Patterns to a Standard Young Tableau.

To find this map:

Let n = i, and use the first row to create the shape of the tableaux (map is given below)**a)**Each element of the 2nd row represents the spaces left empty in each row of the tableaux (on the left side of the row), fill the other spaces of the tableaux with n, let n = (n-1).**b)**Repeat step b, moving down one row (in the GT pattern) each time. Continue this until n = 1. Fill the remaining empty spaces with 1**c)**

This map is given by the Sage command **to_tableau()**. See also this video.

# 9. Sage Reference & Example

# 10. References

*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..

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