Gelfand-Tsetlin Patterns Research Articles

Infinite-Dimensional Representations of the Lie Algebra \mathfrakgl(n,C) Related to Complex Analogs of the Gelfand–Tsetlin Patterns and General Hypergeometric Functions on the Lie Group GL(n,C)

Complex analogs of the Gelfand–Tsetlin patterns are introduced. Infinite-dimensional representations of \(\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)\) in the vector spaces spanned on these patterns are constructed. Exponentials of these representations are described. These exponentials are operators T(x), x∈GL(n,C), defined only in neighborhoods of the identity element of GL(n,C). A system of differential-difference equations for matrix elements of operators T(x) is constructed. Explicit formulas for matrix elements are obtained for the case x∈Z±, where Z+ and Z− are the triangular unipotent subgroups. Representations of \(\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)\) are also constructed; bases of these representations consist of Gelfand–Tsetlin patterns having infinitely many rows.

Gelfand–Tsetlin Pattern and Strict Partitions

We give a representation-theoretical interpretation of the Gelfand–Tsetlin pattern for strict partitions. Using the Howe duality involving a pair of the queer Lie superalgebras and an analogue of the Littlewood–Richardson rule for Schur Q-functions, we show that such patterns give the branching rule for the irreducible tensor representations of the queer Lie superalgebra.

Cones, crystals, and patterns

There are two well known combinatorial tools in the representation theory ofSLn, the semi-standard Young tableaux and the Gelfand-Tsetlin patterns. Using the path model and the theory of crystals, we generalize the concept of patterns to arbitrary complex semi-simple algebraic groups.

A remark on the Gelfand-Tsetlin patterns for symplectic groups

In this paper, we study an approach by Gelfand-Tsetlin to the representation of symplectic groups.