The action of the unitary group on the real Heisenberg group yields a Gelfand pair. The associated spherical functions are well known and have been computed independently by many authors. In this paper we develop a discrete counterpart to this story by replacing the real numbers by a finite field of odd characteristic. This produces a finite Gelfand pair whose spherical functions are computed e...

This paper studies the question of resolvent existence, as well as, smoothness elements from definition domain (separability) a class hyperbolic differential operators defined in an unbounded with greatly increasing coefficients after closure space L2(R2). Such problem was previously put forward by I.M. Gelfand for elliptic operators. Here, we note that detailed analysis shows when studying spe...

This paper is a study of the polyhedral geometry of Gelfand–Tsetlin polytopes arising in the representation theory of glnC and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand–Tsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov...

The Gelfand–Tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups. The boundary of the Gelfand–Tsetlin graph has at least three incarnations — as a discrete potential theory boundary, as the set of finite indecomposable characters of the infinitedimensional unitary group, and as the set of doubly infinite totally positive sequences. An ...

The lattice Gelfand-Dickey hierarchy is a version of the hierarchy. A special case KdV Inspired by recent work Buryak and Rossi, we propose an extension extended system has infinite number logarithmic flows alongside usual flows. We present Lax, Sato Hirota equations factorization problem difference operators that captures whole set solutions. construction this resembles 1D bigraded Toda hierar...

We study the classical version of supersymmetric W-algebras. Using the second Gelfand-Dickey Hamiltonian structure we work out in detail W 2 and W 3-algebras.