The Rocky Mountain Algebraic Combinatorics Seminar (RMAC) will be starting up again this semester with our first talk on February 3, 2023.  Our seminars will happen on some Fridays at 4 PM in Weber 223 on CSU’s campus as usual, with refreshments beforehand at 3:30 in Weber 117.

Gelfand pairs and finite metric spaces
Richard Green
University of Colorado Boulder

A pair of finite groups $(G, H)$ is called a “Gelfand pair” if $H$ is a
subgroup of $G$ and if the trivial character of $H$ is multiplicity free when
induced to $G$. This talk will review some of the basic properties of Gelfand
pairs of finite groups, with particular emphasis on examples that arise
from transitive groups of isometries of finite metric spaces.

A tale of two bases
Richard Green
University of Colorado Boulder

A well-known example of a Gelfand pair $(G, H)$ of finite groups is the
case where $G = S_n$ is the symmetric group, and $H = S_k \times S_{n-k}$
(for $1 \leq k \leq n/2$) is a maximal Young subgroup. We will discuss
two natural bases for the coset space $G/H$, and describe some of their
properties. The first basis is the obvious basis coming from the cosets
themselves, which are naturally indexed by the $k$-element subsets of an
$n$-element set. The second basis, which will be defined in the talk, is
naturally indexed by the set of standard Young tableaux with two rows and
with at most $k$ boxes in the second row.

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