Gelfand-Tsetlin Patterns

AUTHORS:

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

REFERENCES:

[BBF](1, 2, 3) 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.
class sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPattern

Bases: sage.structure.list_clone.ClonableArray

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
Tokuyama_coefficient(name='t')

Return the Tokuyama coefficient attached to self.

Following the exposition of [BBF], Tokuyama’s formula asserts

\[\begin{split}\sum_{G} (t+1)^{s(G)} t^{l(G)} z_1^{d_{n+1}} z_2^{d_{n}-d_{n+1}} \cdots z_{n+1}^{d_1-d_2} = s_{\lambda}(z_1,\dots,z_{n+1}) \prod_{i<j} (z_j+tz_i),\end{split}\]

where the sum is over all strict Gelfand-Tsetlin patterns with fixed top row \(\lambda + \rho\), with \(\lambda\) a partition with at most \(n+1\) parts and \(\rho = (n, n-1, \ldots, 1, 0)\), and \(s_\lambda\) is a Schur function.

INPUT:

  • name – (Default: 't') An alternative name for the variable \(t\).

EXAMPLES:

sage: P = GelfandTsetlinPattern([[3,2,1],[2,2],[2]])
sage: P.Tokuyama_coefficient()
0
sage: G = GelfandTsetlinPattern([[3,2,1],[3,1],[2]])
sage: G.Tokuyama_coefficient()
t^2 + t
sage: G = GelfandTsetlinPattern([[2,1,0],[1,1],[1]])
sage: G.Tokuyama_coefficient()
0
sage: G = GelfandTsetlinPattern([[5,3,2,1,0],[4,3,2,0],[4,2,1],[3,2],[3]])
sage: G.Tokuyama_coefficient()
t^8 + 3*t^7 + 3*t^6 + t^5
boxed_entries()

Return the position of the boxed entries of self.

Using the right-hand rule, an entry \(a_{i,j}\) is boxed if \(a_{i,j} = a_{i-1,j-1}\); i.e., \(a_{i,j}\) has the same value as its neighbor to the northwest.

EXAMPLES:

sage: G = GelfandTsetlinPattern([[3,2,1],[3,1],[1]])
sage: G.boxed_entries()
((1, 0),)
check()

Check that this is a valid Gelfand-Tsetlin pattern.

EXAMPLES:

sage: G = GelfandTsetlinPatterns()
sage: G([[3,2,1],[2,1],[1]]).check()
circled_entries()

Return the circled entries of self.

Using the right-hand rule, an entry \(a_{i,j}\) is circled if \(a_{i,j} = a_{i-1,j}\); i.e., \(a_{i,j}\) has the same value as its neighbor to the northeast.

EXAMPLES:

sage: G = GelfandTsetlinPattern([[3,2,1],[3,1],[1]])
sage: G.circled_entries()
((1, 1), (2, 0))
is_strict()

Return True if self is a strict Gelfand-Tsetlin pattern.

A Gelfand-Tsetlin pattern is said to be strict if every row is strictly decreasing.

EXAMPLES:

sage: GelfandTsetlinPattern([[7,3,1],[6,2],[4]]).is_strict()
True
sage: GelfandTsetlinPattern([[3,2,1],[3,1],[1]]).is_strict()
True
sage: GelfandTsetlinPattern([[6,0,0],[3,0],[2]]).is_strict()
False
number_of_boxes()

Return the number of boxed entries. See boxed_entries().

EXAMPLES:

sage: G = GelfandTsetlinPattern([[3,2,1],[3,1],[1]])
sage: G.number_of_boxes()
1
number_of_circles()

Return the number of boxed entries. See circled_entries().

EXAMPLES:

sage: G = GelfandTsetlinPattern([[3,2,1],[3,1],[1]])
sage: G.number_of_circles()
2
number_of_special_entries()

Return the number of special entries. See special_entries().

EXAMPLES:

sage: G = GelfandTsetlinPattern([[4,2,1],[4,1],[2]])
sage: G.number_of_special_entries()
1
pp()

Pretty print self.

EXAMPLES:

sage: G = GelfandTsetlinPatterns()
sage: G([[3,2,1],[2,1],[1]]).pp()
  3     2     1
     2     1
        1
row_sums()

Return the list of row sums.

For a Gelfand-Tsetlin pattern \(G\), the \(i\)-th row sum \(d_i\) is

\[d_i = d_i(G) = \sum_{j=i}^{n} a_{i,j}.\]

EXAMPLES:

sage: G = GelfandTsetlinPattern([[5,3,2,1,0],[4,3,2,0],[4,2,1],[3,2],[3]])
sage: G.row_sums()
[11, 9, 7, 5, 3]
sage: G = GelfandTsetlinPattern([[3,2,1],[3,1],[2]])
sage: G.row_sums()
[6, 4, 2]
special_entries()

Return the special entries.

An entry \(a_{i,j}\) is special if \(a_{i-1,j-1} > a_{i,j} > a_{i-1,j}\), that is to say, the entry is neither boxed nor circled and is not in the first row. The name was coined by [Tok88].

EXAMPLES:

sage: G = GelfandTsetlinPattern([[3,2,1],[3,1],[1]])
sage: G.special_entries()
()
sage: G = GelfandTsetlinPattern([[4,2,1],[4,1],[2]])
sage: G.special_entries()
((2, 0),)
to_tableau(*args, **kwds)

Return self as a semistandard Young tableau.

The conversion from a Gelfand-Tsetlin pattern to a semistandard Young tableaux is as follows. 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 \(k=1,\dots,n\).

EXAMPLES:

sage: G = GelfandTsetlinPatterns()
sage: elt = G([[3,2,1],[2,1],[1]])
sage: T = elt.to_tableau(); T
[[1, 2, 3], [2, 3], [3]]
sage: T.pp()
  1  2  3
  2  3
  3
sage: G(T) == elt
True
weight()

Return the weight of self.

Define the weight of \(G\) to be the content of the tableau to which \(G\) corresponds under the bijection between Gelfand-Tsetlin patterns and semistandard tableaux. More precisely,

\[\mathrm{wt}(G) = (d_n, d_{n-1}-d_n, \dots, d_1-d_2),\]

where the \(d_i\) are the row sums.

EXAMPLES:

sage: G = GelfandTsetlinPattern([[2,1,0],[1,0],[1]])
sage: G.weight()
(1, 0, 2)
sage: G = GelfandTsetlinPattern([[4,2,1],[3,1],[2]])
sage: G.weight()
(2, 2, 3)
class sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPatterns(n, k, strict)

Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation

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

TESTS:

Check that the number of Gelfand-Tsetlin patterns is equal to the number of semistandard Young tableaux:

sage: G = GelfandTsetlinPatterns(3,3)
sage: c = 0
sage: from sage.combinat.crystals.kirillov_reshetikhin import partitions_in_box
sage: for p in partitions_in_box(3,3):
...      S = SemistandardTableaux(p, max_entry=3)
...      c += S.cardinality()
sage: c == G.cardinality()
True

Note that the top row in reverse of the Gelfand-Tsetlin pattern is the shape of the corresponding semistandard Young tableau under the bijection described in GelfandTsetlinPattern.to_tableau():

sage: G = GelfandTsetlinPatterns(top_row=[2,2,1])
sage: S = SemistandardTableaux([2,2,1], max_entry=3)
sage: G.cardinality() == S.cardinality()
True
Element

alias of GelfandTsetlinPattern

class sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPatternsTopRow(top_row, strict)

Bases: sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPatterns

Gelfand-Tsetlin patterns with a fixed top row.

Tokuyama_formula(name='t')

Return the Tokuyama formula of self.

Following the exposition of [BBF], Tokuyama’s formula asserts

\[\begin{split}\sum_{G} (t+1)^{s(G)} t^{l(G)} z_1^{d_{n+1}} z_2^{d_{n}-d_{n+1}} \cdots z_{n+1}^{d_1-d_2} = s_{\lambda} (z_1, \ldots, z_{n+1}) \prod_{i<j} (z_j+tz_i),\end{split}\]

where the sum is over all strict Gelfand-Tsetlin patterns with fixed top row \(\lambda+\rho\), with \(\lambda\) a partition with at most \(n+1\) parts and \(\rho = (n,n-1,\dots,1,0)\), and \(s_{\lambda}\) is a Schur function.

INPUT:

  • name – (Default: 't') An alternative name for the variable \(t\).

EXAMPLES:

sage: GT = GelfandTsetlinPatterns(top_row=[2,1,0],strict=True)
sage: GT.Tokuyama_formula()
t^3*x1^2*x2 + t^2*x1*x2^2 + t^2*x1^2*x3 + t^2*x1*x2*x3 + t*x1*x2*x3 + t*x2^2*x3 + t*x1*x3^2 + x2*x3^2
sage: GT = GelfandTsetlinPatterns(top_row=[3,2,1],strict=True)
sage: GT.Tokuyama_formula()
t^3*x1^3*x2^2*x3 + t^2*x1^2*x2^3*x3 + t^2*x1^3*x2*x3^2 + t^2*x1^2*x2^2*x3^2 + t*x1^2*x2^2*x3^2 + t*x1*x2^3*x3^2 + t*x1^2*x2*x3^3 + x1*x2^2*x3^3
sage: GT = GelfandTsetlinPatterns(top_row=[1,1,1],strict=True)
sage: GT.Tokuyama_formula()
0
top_row()

Return the top row of self.

EXAMPLES:

sage: G = GelfandTsetlinPatterns(top_row=[4,4,3,1])
sage: G.top_row()
(4, 4, 3, 1)

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