The generating function of strict Gelfand patterns and some formulas on characters of general linear groups. (English) Zbl 0639.20022
Introducing a new generating function of strict Gelfand patterns, we give a formula which can be specialized to the following three important formulas in representation theory: Gelfand-Tsetlin parametrization, Stanley’s formula, and Weyl’s character formula.
Gelfand patterns are triangular arrays of non-negative integers satisfying certain conditions. I. M. Gelfand and M. L. Tsetlin introduced them for the parametrization of the weight vectors of representation spaces of unitary groups [Dokl. Akad. Nauk SSSR 71, 825–828 (1950; Zbl 0037.15301)]. R. P. Stanley’s formula [Lect. Notes Math. 1234, 285–293 (1986; Zbl 0612.05008)] relates a generating function of strict Gelfand patterns with the ‘most singular’ values of the Hall-Littlewood polynomials, which is a fundamental tool for investigating the representations of general linear groups over finite fields and local fields.
Our formula is the following. Let \(\lambda\) be a partition of length \(n\), and \(\alpha =\lambda +\delta\) be its canonically corresponding distinct partition.
Theorem. \[ J(\alpha;t;z_ 1,z_ 2,...,z_{n-1},z_ n)=\left\{\prod _{j>i}(z_ i+z_ jt)\right\}\, S_{\lambda}(z_ 1,z_ 2,...,z_{n-1},z_ n) \] where \(J\) is the generating function of strict Gelfand patterns defined in the paper and \(S_{\lambda}\) is the Schur function associated with the rational representation of \(\text{GL}(n,\mathbb C)\) with highest weight \(\lambda\).
Substituting \(0, 1, -1\) for the parameter \(t\) in our theorem, we get the three classical formulas mentioned above. Besides, as a corollary, we have a neat generalization of Weyl’s denominator formula of general linear groups. “Alternating sign matrices” is used as a naturally extended notion of permutations there.
Gelfand patterns are triangular arrays of non-negative integers satisfying certain conditions. I. M. Gelfand and M. L. Tsetlin introduced them for the parametrization of the weight vectors of representation spaces of unitary groups [Dokl. Akad. Nauk SSSR 71, 825–828 (1950; Zbl 0037.15301)]. R. P. Stanley’s formula [Lect. Notes Math. 1234, 285–293 (1986; Zbl 0612.05008)] relates a generating function of strict Gelfand patterns with the ‘most singular’ values of the Hall-Littlewood polynomials, which is a fundamental tool for investigating the representations of general linear groups over finite fields and local fields.
Our formula is the following. Let \(\lambda\) be a partition of length \(n\), and \(\alpha =\lambda +\delta\) be its canonically corresponding distinct partition.
Theorem. \[ J(\alpha;t;z_ 1,z_ 2,...,z_{n-1},z_ n)=\left\{\prod _{j>i}(z_ i+z_ jt)\right\}\, S_{\lambda}(z_ 1,z_ 2,...,z_{n-1},z_ n) \] where \(J\) is the generating function of strict Gelfand patterns defined in the paper and \(S_{\lambda}\) is the Schur function associated with the rational representation of \(\text{GL}(n,\mathbb C)\) with highest weight \(\lambda\).
Substituting \(0, 1, -1\) for the parameter \(t\) in our theorem, we get the three classical formulas mentioned above. Besides, as a corollary, we have a neat generalization of Weyl’s denominator formula of general linear groups. “Alternating sign matrices” is used as a naturally extended notion of permutations there.
Reviewer: T. Tokuyama
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 20C30 | Representations of finite symmetric groups |
| 22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
| 05E10 | Combinatorial aspects of representation theory |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
