Representation Theory Of Classical Groups Research Articles

Branching laws for classical groups: the non-tempered case

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

Hibi Algebras and Representation Theory

This paper gives a survey on the relation between Hibi algebras and representation theory. The notion of Hodge algebras or algebras with straightening laws has been proved to be very useful to describe the structure of many important algebras in classical invariant theory and representation theory (Bruns and Herzog 1993; De Concini et al. 1982; Eisenbud 1980; Gonciulea and Lakshmibai 2001; Seshadri 2007). In particular, a special type of such algebras introduced by Hibi (1987) provides a nice bridge between combinatorics and representation theory of classical groups. We will examine certain poset structures of Young tableaux and affine monoids, Hibi algebras in toric degenerations of flag varieties, and their relations to polynomial representations of the complex general linear group.

Random surface growth and Karlin-McGregor polynomials

We consider consistent dynamics for non-intersecting birth and death chains, originating from dualities of stochastic coalescing flows and one dimensional orthogonal polynomials. As corollaries, we obtain unified and simple probabilistic proofs of certain key intertwining relations between multivariate Markov chains on the levels of some branching graphs. Special cases include the dynamics on the Gelfand-Tsetlin graph considered in the seminal work of Borodin and Olshanski in [10] and the ones on the BC-type graph recently studied by Cuenca in [17]. Moreover, we introduce a general inhomogeneous random growth process with a wall that includes as special cases the ones considered by Borodin and Kuan [8] and Cerenzia [15], that are related to the representation theory of classical groups and also the Jacobi growth process more recently studied by Cerenzia and Kuan [16]. Its most important feature is that, this process retains the determinantal structure of the ones studied previously and for the fully packed initial condition we are able to calculate its correlation kernel explicitly in terms of a contour integral involving orthogonal polynomials. At a certain scaling limit, at a finite distance from the wall, one obtains for a single level discrete determinantal ensembles associated to continuous orthogonal polynomials, that were recently introduced by Borodin and Olshanski in [11], and that depend on the inhomogeneities.

Noetherianity of some degree two twisted commutative algebras

In recent years, researchers have discovered various large algebraic structures that have surprising finiteness properties, such as FI-modules and Delta-modules. In this paper, we add another example to the growing list: we show that certain degree two twisted commutative algebras are noetherian. This example appears to have some fundamental differences from previous examples, and is therefore especially interesting. Reflective of this, our proof introduces new methods for establishing noetherianity that are likely to be applicable in other situations. The algebras considered in this paper are closely related to the stable representation theory of classical groups, which is one source of motivation for their study.

Categories parametrized by schemes and representation theory in complex rank

Many key invariants in the representation theory of classical groups (symmetric groups Sn, matrix groups GLn, On, Sp2n) are polynomials in n (e.g., dimensions of irreducible representations). This allowed Deligne to extend the representation theory of these groups to complex values of the rank n. Namely, Deligne defined generically semisimple families of tensor categories parametrized by n∈C, which at positive integer n specialize to the classical representation categories. Using Deligneʼs work, Etingof proposed a similar extrapolation for many non-semisimple representation categories built on representation categories of classical groups, e.g., degenerate affine Hecke algebras (dAHA). It is expected that for generic n∈C such extrapolations behave as they do for large integer n (“stabilization”).The goal of our work is to provide a technique to prove such statements. Namely, we develop an algebro-geometric framework to study categories indexed by a parameter n, in which the set of values of n for which the category has a given property is constructible. This implies that if a property holds for integer n, it then holds for generic complex n. We use this to give a new proof that Deligneʼs categories are generically semisimple. We also apply this method to Etingofʼs extrapolations of dAHA, and prove that when n is transcendental, “finite-dimensional” simple objects are quotients of certain standard induced objects, extrapolating Zelevinskyʼs classification of simple dAHA-modules for n∈N. Finally, we obtain similar results for the extrapolations of categories associated to wreath products of the symmetric group with associative algebras.