Events

Title: The degrees of Stiefel Manifolds

Abstract:

The set of orthonormal bases for k-planes in R^n is cut out by the equations X*X^T = I
where X is a k x n matrix of variables and I is k x k identity. This space, known as the Stiefel manifold St(k,n), generalizes the orthogonal group and can be realized as the homogeneous space O(n)/O(n-k). Its algebraic closure
gives a complex affine variety, and thus, it has a degree.

I will discuss our derivation of these degrees. Extending 2017 work on the degrees of special orthogonal groups, joint work with Fulvio Gesmundo gives a combinatorial formula in terms of non-intersecting lattice paths.
This result relies on representation theory, commutative algebra, Ehrhart theory, polyhedral geometry, and enumerative combinatorics.

I will conclude with some open problems inspired by these objects.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm.

Abstract: In this talk, we consider a linear algebraic object known as a tridiagonal pair which arises naturally in the context of Q-polynomial distance-regular graphs. We will focus on a special class of tridiagonal pairs said to have Racah type. Given a tridiagonal pair of Racah type, we associate with it several linear transformations which act on the underlying vector space in an attractive manner and discuss their relationships with one another. In an earlier work, we introduced the double lowering operator Ψ for a tridiagonal pair. In this talk, we will explore this double lowering map further under the assumption that our tridiagonal pair has Racah type and will use the double lowering map to obtain new relations involving the operators associated with two oriented versions of our tridiagonal pair.