Gelfand-Tsetlin Patterns

AUTHORS:

  • Travis Scrimshaw (2013-15-03): Initial version

REFERENCES:

[BBF]B. Brubaker, D. Bump, and S. Friedberg. Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory. Ann. of Math. Stud., vol. 175, Princeton Univ. Press, New Jersey, 2011.
[GC50]I. M. Gelfand and M. L. Cetlin. Finite-Dimensional Representations of the Group of Unimodular Matrices. Dokl. Akad. Nauk SSSR 71, pp. 825–828, 1950.
[Tok88]T. Tokuyama. A Generating Function of Strict Gelfand Patterns and Some Formulas on Characters of General Linear Groups. J. Math. Soc. Japan 40 (4), pp. 671–685, 1988.
sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPattern

A Gelfand-Tsetlin (sometimes written as Gelfand-Zetlin or Gelfand-Cetlin) pattern. They were originally defined in [GC50].

A Gelfand-Tsetlin pattern is a triangular array:

\[\begin{split}\begin{array}{ccccccccc} a_{1,1} & & a_{1,2} & & a_{1,3} & & \cdots & & a_{1,n} \\ & a_{2,2} & & a_{2,3} & & \cdots & & a_{2,n} \\ & & a_{3,3} & & \cdots & & a_{3,n} \\ & & & \ddots \\ & & & & a_{n,n} \end{array}\end{split}\]

such that \(a_{i,j} \geq a_{i+1,j+1} \geq a_{i,j+1}\).

Gelfand-Tsetlin patterns are in bijection with semistandard Young tableaux by the following algorithm. Let \(G\) be a Gelfand-Tsetlin pattern with \(\lambda^{(k)}\) being the \((n-k+1)\)-st row (note that this is a partition). The definition of \(G\) implies

\[\lambda^{(0)} \subseteq \lambda^{(1)} \subseteq \cdots \subseteq \lambda^{(n)},\]

where \(\lambda^{(0)}\) is the empty partition, and each skew shape \(\lambda^{(k)}/\lambda^{(k-1)}\) is a horizontal strip. Thus define \(T(G)\) by inserting \(k\) into the squares of the skew shape \(\lambda^{(k)}/ \lambda^{(k-1)}\), for \(k=1,\dots,n\).

To each entry in a Gelfand-Tsetlin pattern, one may attach a decoration of a circle or a box (or both or neither). These decorations appear in the study of Weyl group multiple Dirichlet series, and are implemented here following the exposition in [BBF].

Note

We use the “right-hand” rule for determining circled and boxed entries.

Warning

The entries in Sage are 0-based and are thought of as flushed to the left in a matrix. In other words, the coordinates of entries in the Gelfand-Tsetlin patterns are thought of as the matrix:

\[\begin{split}\begin{bmatrix} g_{0,0} & g_{0,1} & g_{0,2} & \cdots & g_{0,n-2} & g_{n-1,n-1} \\ g_{1,0} & g_{1,1} & g_{1,2} & \cdots & g_{1,n-2} \\ g_{2,0} & g_{2,1} & g_{2,2} & \cdots \\ \vdots & \vdots & \vdots \\ g_{n-2,0} & g_{n-2,1} \\ g_{n-1,0} \end{bmatrix}.\end{split}\]

However, in the discussions, we will be using the standard numbering system.

EXAMPLES:

sage: G = GelfandTsetlinPattern([[3, 2, 1], [2, 1], [1]]); G
[[3, 2, 1], [2, 1], [1]]
sage: G.pp()
  3     2     1
     2     1
        1
sage: G = GelfandTsetlinPattern([[7, 7, 4, 0], [7, 7, 3], [7, 5], [5]]); G.pp()
  7     7     4     0
     7     7     3
        7     5
           5
sage: G.to_tableau().pp()
  1  1  1  1  1  2  2
  2  2  2  2  2  3  3
  3  3  3  4
sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPatterns

Gelfand-Tsetlin patterns.

INPUT:

  • n – The width or depth of the array, also known as the rank
  • k – (Default: None) If specified, this is the maximum value that can occur in the patterns
  • top_row – (Default: None) If specified, this is the fixed top row of all patterns
  • strict – (Default: False) Set to True if all patterns are strict patterns
sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPatternsTopRow

Gelfand-Tsetlin patterns with a fixed top row.