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. |
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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
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sage.combinat.gelfand_tsetlin_patterns.
GelfandTsetlinPatterns
¶ Gelfand-Tsetlin patterns.
INPUT:
n
– The width or depth of the array, also known as the rankk
– (Default:None
) If specified, this is the maximum value that can occur in the patternstop_row
– (Default:None
) If specified, this is the fixed top row of all patternsstrict
– (Default:False
) Set toTrue
if all patterns are strict patterns
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sage.combinat.gelfand_tsetlin_patterns.
GelfandTsetlinPatternsTopRow
¶ Gelfand-Tsetlin patterns with a fixed top row.