Gelfand-Tsetlin Patterns¶
AUTHORS:
- Travis Scrimshaw (2013-15-03): Initial version 
REFERENCES:
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.
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.
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[source]¶
- Bases: - 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 - >>> from sage.all import * >>> G = GelfandTsetlinPattern([[Integer(3), Integer(2), Integer(1)], [Integer(2), Integer(1)], [Integer(1)]]); G [[3, 2, 1], [2, 1], [1]] >>> G.pp() 3 2 1 2 1 1 >>> G = GelfandTsetlinPattern([[Integer(7), Integer(7), Integer(4), Integer(0)], [Integer(7), Integer(7), Integer(3)], [Integer(7), Integer(5)], [Integer(5)]]); G.pp() 7 7 4 0 7 7 3 7 5 5 >>> 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')[source]¶
- Return the Tokuyama coefficient attached to - self.- Following the exposition of [BBF], Tokuyama’s formula asserts \[\sum_{G} (t+1)^{s(G)} t^{l(G)} z_1^{d_{n+1}} z_2^{d_{n}-d_{n+1}} \cdots z_{n+1}^{d_1-d_2} = s_{\lambda}(z_1,\dots,z_{n+1}) \prod_{i<j} (z_j+tz_i),\]- 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 - >>> from sage.all import * >>> P = GelfandTsetlinPattern([[Integer(3),Integer(2),Integer(1)],[Integer(2),Integer(2)],[Integer(2)]]) >>> P.Tokuyama_coefficient() 0 >>> G = GelfandTsetlinPattern([[Integer(3),Integer(2),Integer(1)],[Integer(3),Integer(1)],[Integer(2)]]) >>> G.Tokuyama_coefficient() t^2 + t >>> G = GelfandTsetlinPattern([[Integer(2),Integer(1),Integer(0)],[Integer(1),Integer(1)],[Integer(1)]]) >>> G.Tokuyama_coefficient() 0 >>> G = GelfandTsetlinPattern([[Integer(5),Integer(3),Integer(2),Integer(1),Integer(0)],[Integer(4),Integer(3),Integer(2),Integer(0)],[Integer(4),Integer(2),Integer(1)],[Integer(3),Integer(2)],[Integer(3)]]) >>> G.Tokuyama_coefficient() t^8 + 3*t^7 + 3*t^6 + t^5 
 - bender_knuth_involution(i)[source]¶
- Return the image of - selfunder the \(i\)-th Bender-Knuth involution.- If the triangle - selfhas size \(n\) then this is defined for \(0 < i < n\).- The entries of - selfcan take values in any ordered ring. Usually, this will be the integers but can also be the rationals or the real numbers.- This implements the construction of the Bender-Knuth involution using toggling due to Berenstein-Kirillov. - This agrees with the Bender-Knuth involution on semistandard tableaux. - EXAMPLES: - sage: G = GelfandTsetlinPattern([[5,3,2,1,0],[4,3,2,0],[4,2,1],[3,2],[3]]) sage: G.bender_knuth_involution(2) [[5, 3, 2, 1, 0], [4, 3, 2, 0], [4, 2, 1], [4, 1], [3]] sage: G = GelfandTsetlinPattern([[3,2,0],[2.2,0],[2]]) sage: G.bender_knuth_involution(2) [[3, 2, 0], [2.80000000000000, 2], [2]] - >>> from sage.all import * >>> G = GelfandTsetlinPattern([[Integer(5),Integer(3),Integer(2),Integer(1),Integer(0)],[Integer(4),Integer(3),Integer(2),Integer(0)],[Integer(4),Integer(2),Integer(1)],[Integer(3),Integer(2)],[Integer(3)]]) >>> G.bender_knuth_involution(Integer(2)) [[5, 3, 2, 1, 0], [4, 3, 2, 0], [4, 2, 1], [4, 1], [3]] >>> G = GelfandTsetlinPattern([[Integer(3),Integer(2),Integer(0)],[RealNumber('2.2'),Integer(0)],[Integer(2)]]) >>> G.bender_knuth_involution(Integer(2)) [[3, 2, 0], [2.80000000000000, 2], [2]] 
 - boxed_entries()[source]¶
- 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),) - >>> from sage.all import * >>> G = GelfandTsetlinPattern([[Integer(3),Integer(2),Integer(1)],[Integer(3),Integer(1)],[Integer(1)]]) >>> G.boxed_entries() ((1, 0),) 
 - check()[source]¶
- Check that this is a valid Gelfand-Tsetlin pattern. - EXAMPLES: - sage: G = GelfandTsetlinPatterns() sage: G([[3,2,1],[2,1],[1]]).check() - >>> from sage.all import * >>> G = GelfandTsetlinPatterns() >>> G([[Integer(3),Integer(2),Integer(1)],[Integer(2),Integer(1)],[Integer(1)]]).check() 
 - circled_entries()[source]¶
- 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)) - >>> from sage.all import * >>> G = GelfandTsetlinPattern([[Integer(3),Integer(2),Integer(1)],[Integer(3),Integer(1)],[Integer(1)]]) >>> G.circled_entries() ((1, 1), (2, 0)) 
 - is_strict()[source]¶
- Return - Trueif- selfis 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 - >>> from sage.all import * >>> GelfandTsetlinPattern([[Integer(7),Integer(3),Integer(1)],[Integer(6),Integer(2)],[Integer(4)]]).is_strict() True >>> GelfandTsetlinPattern([[Integer(3),Integer(2),Integer(1)],[Integer(3),Integer(1)],[Integer(1)]]).is_strict() True >>> GelfandTsetlinPattern([[Integer(6),Integer(0),Integer(0)],[Integer(3),Integer(0)],[Integer(2)]]).is_strict() False 
 - number_of_boxes()[source]¶
- 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 - >>> from sage.all import * >>> G = GelfandTsetlinPattern([[Integer(3),Integer(2),Integer(1)],[Integer(3),Integer(1)],[Integer(1)]]) >>> G.number_of_boxes() 1 
 - number_of_circles()[source]¶
- 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 - >>> from sage.all import * >>> G = GelfandTsetlinPattern([[Integer(3),Integer(2),Integer(1)],[Integer(3),Integer(1)],[Integer(1)]]) >>> G.number_of_circles() 2 
 - number_of_special_entries()[source]¶
- 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 - >>> from sage.all import * >>> G = GelfandTsetlinPattern([[Integer(4),Integer(2),Integer(1)],[Integer(4),Integer(1)],[Integer(2)]]) >>> G.number_of_special_entries() 1 
 - pp()[source]¶
- Pretty print - self.- EXAMPLES: - sage: G = GelfandTsetlinPatterns() sage: G([[3,2,1],[2,1],[1]]).pp() 3 2 1 2 1 1 - >>> from sage.all import * >>> G = GelfandTsetlinPatterns() >>> G([[Integer(3),Integer(2),Integer(1)],[Integer(2),Integer(1)],[Integer(1)]]).pp() 3 2 1 2 1 1 
 - row_sums()[source]¶
- 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] - >>> from sage.all import * >>> G = GelfandTsetlinPattern([[Integer(5),Integer(3),Integer(2),Integer(1),Integer(0)],[Integer(4),Integer(3),Integer(2),Integer(0)],[Integer(4),Integer(2),Integer(1)],[Integer(3),Integer(2)],[Integer(3)]]) >>> G.row_sums() [11, 9, 7, 5, 3] >>> G = GelfandTsetlinPattern([[Integer(3),Integer(2),Integer(1)],[Integer(3),Integer(1)],[Integer(2)]]) >>> G.row_sums() [6, 4, 2] 
 - special_entries()[source]¶
- 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),) - >>> from sage.all import * >>> G = GelfandTsetlinPattern([[Integer(3),Integer(2),Integer(1)],[Integer(3),Integer(1)],[Integer(1)]]) >>> G.special_entries() () >>> G = GelfandTsetlinPattern([[Integer(4),Integer(2),Integer(1)],[Integer(4),Integer(1)],[Integer(2)]]) >>> G.special_entries() ((2, 0),) 
 - to_tableau()[source]¶
- Return - selfas 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 - >>> from sage.all import * >>> G = GelfandTsetlinPatterns() >>> elt = G([[Integer(3),Integer(2),Integer(1)],[Integer(2),Integer(1)],[Integer(1)]]) >>> T = elt.to_tableau(); T [[1, 2, 3], [2, 3], [3]] >>> T.pp() 1 2 3 2 3 3 >>> G(T) == elt True 
 - weight()[source]¶
- 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) - >>> from sage.all import * >>> G = GelfandTsetlinPattern([[Integer(2),Integer(1),Integer(0)],[Integer(1),Integer(0)],[Integer(1)]]) >>> G.weight() (1, 0, 2) >>> G = GelfandTsetlinPattern([[Integer(4),Integer(2),Integer(1)],[Integer(3),Integer(1)],[Integer(2)]]) >>> G.weight() (2, 2, 3) 
 
- class sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPatterns(n, k, strict)[source]¶
- Bases: - UniqueRepresentation,- Parent- 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- Trueif all patterns are strict patterns
 - Element[source]¶
- alias of - GelfandTsetlinPattern
 - random_element()[source]¶
- Return a uniformly random Gelfand-Tsetlin pattern. - EXAMPLES: - sage: g = GelfandTsetlinPatterns(4, 5) sage: x = g.random_element() sage: x in g True sage: len(x) 4 sage: all(y in range(5+1) for z in x for y in z) True sage: x.check() - >>> from sage.all import * >>> g = GelfandTsetlinPatterns(Integer(4), Integer(5)) >>> x = g.random_element() >>> x in g True >>> len(x) 4 >>> all(y in range(Integer(5)+Integer(1)) for z in x for y in z) True >>> x.check() - sage: g = GelfandTsetlinPatterns(4, 5, strict=True) sage: x = g.random_element() sage: x in g True sage: len(x) 4 sage: all(y in range(5+1) for z in x for y in z) True sage: x.check() sage: x.is_strict() True - >>> from sage.all import * >>> g = GelfandTsetlinPatterns(Integer(4), Integer(5), strict=True) >>> x = g.random_element() >>> x in g True >>> len(x) 4 >>> all(y in range(Integer(5)+Integer(1)) for z in x for y in z) True >>> x.check() >>> x.is_strict() True 
 
- class sage.combinat.gelfand_tsetlin_patterns.GelfandTsetlinPatternsTopRow(top_row, strict)[source]¶
- Bases: - GelfandTsetlinPatterns- Gelfand-Tsetlin patterns with a fixed top row. - Tokuyama_formula(name='t')[source]¶
- Return the Tokuyama formula of - self.- Following the exposition of [BBF], Tokuyama’s formula asserts \[\sum_{G} (t+1)^{s(G)} t^{l(G)} z_1^{d_{n+1}} z_2^{d_{n}-d_{n+1}} \cdots z_{n+1}^{d_1-d_2} = s_{\lambda} (z_1, \ldots, z_{n+1}) \prod_{i<j} (z_j+tz_i),\]- 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 - >>> from sage.all import * >>> GT = GelfandTsetlinPatterns(top_row=[Integer(2),Integer(1),Integer(0)],strict=True) >>> 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 >>> GT = GelfandTsetlinPatterns(top_row=[Integer(3),Integer(2),Integer(1)],strict=True) >>> 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 >>> GT = GelfandTsetlinPatterns(top_row=[Integer(1),Integer(1),Integer(1)],strict=True) >>> GT.Tokuyama_formula() 0 
 - random_element()[source]¶
- Return a uniformly random Gelfand-Tsetlin pattern with specified top row. - EXAMPLES: - sage: g = GelfandTsetlinPatterns(top_row = [4, 3, 1, 1]) sage: x = g.random_element() sage: x in g True sage: x[0] == [4, 3, 1, 1] True sage: x.check() sage: g = GelfandTsetlinPatterns(top_row=[4, 3, 2, 1], strict=True) sage: x = g.random_element() sage: x in g True sage: x[0] == [4, 3, 2, 1] True sage: x.is_strict() True sage: x.check() - >>> from sage.all import * >>> g = GelfandTsetlinPatterns(top_row = [Integer(4), Integer(3), Integer(1), Integer(1)]) >>> x = g.random_element() >>> x in g True >>> x[Integer(0)] == [Integer(4), Integer(3), Integer(1), Integer(1)] True >>> x.check() >>> g = GelfandTsetlinPatterns(top_row=[Integer(4), Integer(3), Integer(2), Integer(1)], strict=True) >>> x = g.random_element() >>> x in g True >>> x[Integer(0)] == [Integer(4), Integer(3), Integer(2), Integer(1)] True >>> x.is_strict() True >>> x.check()