Dissertation
Inflated Weight, a Dual Approach to Structure Constants for K–Theory of Grassmannians, and a Charge Statistic for Shifted Tableaux
ProQuest Dissertations Publishing
01 Jan 2018
Abstract
The problem of computing products of Schubert classes in the cohomology ring can be formulated as the problem of expanding skew Schur polynomials into the basis of ordinary Schur polynomials. In contrast, the problem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to its basis of Schubert structure sheaves is not equivalent to expanding skew stable Grothendieck polynomials into the basis of ordinary stable Grothendiecks. Instead, we show that the appropriate K-theoretic analogy is through the expansion of skew reverse plane partitions into the basis of polynomials which are Hopf-dual to stable Grothendieck polynomials. We combinatorially prove this expansion is determined by Yamanouchi set-valued tableaux. A by-product of our results is a dual approach proof for Buch’s K-theoretic Littlewood-Richardson rule for the product of stable Grothendieck polynomials. Additionally we answer a question posed by Wan and Wang in 2011. The Kostka-Foulkes polynomials are ubiquitous in algebraic combinatorics, geometry, and representation theory. These polynomials in a parameter t with positive coefficients have deep connections to the Springer theory of Weyl group representations and Lusztig's q-weight multiplicity in finite dimensional irreducible representations of the general linear Lie algebra, among other things. Lascoux and Schutzenberger provided an elegant description of the Kostka-Foulkes polynomials in terms of charge, a statistic associated to Young tableaux. Wan and Wang introduced a spin counterpart, the spin Kostka polynomials, and asked that they be given a Lascoux and Schutzenberger type description by a statistic on shifted Young tableaux. The spin charge statistic given here fulfills this request.
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Details
- Title
- Inflated Weight, a Dual Approach to Structure Constants for K–Theory of Grassmannians, and a Charge Statistic for Shifted Tableaux
- Creators
- Patrick Shields
- Publisher
- ProQuest Dissertations Publishing
- Resource Type
- Dissertation
- Language
- English
- Identifiers
- 991019173762104721