Sangjib Kim

Distributive lattices, affine semigroups, and branching rules of the classical groups

We study algebras encoding stable range branching rules for the pairs of complex classical groups of the same type in the context of toric degenerations of spherical varieties. By lifting affine semigroup algebras constructed from combinatorial data of branching multiplicities, we obtain algebras having highest weight vectors in multiplicity spaces as their standard monomial type bases. In particular, we identify a family of distributive lattices and their associated Hibi algebras which can uniformly describe the stable range branching algebras for all the pairs we consider.

ROW CONVEX TABLEAUX AND BOTT–SAMELSON VARIETIES

By using row convex tableaux, we study the section rings of Bott-Samelson varieties of type A. We obtain flat deformations and standard monomial type bases of the section rings. In a separate section, we investigate a three dimensional Bott-Samelson variety in detail and compute its Hilbert polynomial and toric degenerations.

The nullcone in the multi-vector representation of the symplectic group and related combinatorics

We study the nullcone in the multi-vector representation of the symplectic group with respect to a joint action of the general linear group and the symplectic group. By extracting an algebra over a distributive lattice structure from the coordinate ring of the nullcone, we describe a toric degeneration and standard monomial theory of the nullcone in terms of double tableaux and integral points in a convex polyhedral cone.

Double Pieri algebras and iterated Pieri algebras for the classical groups

We study iterated Pieri rules for representations of classical groups. That is, we consider tensor products of a general representation with multiple factors of representations corresponding to one-rowed Young diagrams (or in the case of the general linear group, also the duals of these). We define {\it iterated Pieri algebras}, whose structure encodes the irreducible decompositions of such tensor products. We show that there is a single family of algebras, which we call {\it double Pieri algebras}, and which can be used to describe the iterated Pieri algebras for all three families of classical groups. Furthermore, we show that the double Pieri algebras have flat deformations to Hibi rings on explicitly described posets. As an interesting application, we describe the branching rules for certain unitary highest weight modules.

TILING BRANCHING MULTIPLICITY SPACES WITH GL PATTERN BLOCKS

We study branching multiplicity spaces of complex classical groups in terms of GL(2) representations. In particular, we show how combinatorics of GL(2) representations are intertwined to make branching rules under the restriction of GL(n) to GL(n-2). We also discuss analogous results for the symplectic and orthogonal groups.

Standard Monomial Theory for Harmonics in Classical Invariant Theory

In this paper, we establish a standard monomial theory, analogous to that of Hodge for GLn, for the harmonic polynomials of a classical action in the sense of Weyl.

Standard Monomial Theory of RR Varieties

We construct the RR varieties as the fiber products of Bott-Samelson varieties over Richardson varieties. We study their homogeneous coordinate rings and standard monomial theory.

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.

On Kostant's partial order on hyperbolic elements

We study Kostants partial order on the elements of a semisimple Lie group in relations with the finite-dimensional representations. In particular, we prove the converse statement of Theorem 6.1 given by Kostant [B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. 6 (1973), pp. 413–455] on hyperbolic elements.