Parabolic Kazhdan-Lusztig polynomials, plethysm and gereralized Hall-Littlewood functions for classical types

We use power sums plethysm operators to introduce H functions which interpolate between the Weyl characters and the Hall-Littlewood functions Q' corresponding to classical Lie groups. The coefficients of these functions on the basis of Weyl characters are parabolic Kazhdan-Lusztig polynomials and thus, are nonnegative. We prove that they can be regarded as quantizations of branching coefficients obtained by restriction to certain Levi subgroups. The H functions associated to linear groups coincide with the polynomials introduced by Lascoux Leclerc and Thibon (LLT polynomials).Comment: To appear in European Journal of Combinatoric

Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials

We show that the Littlewood-Richardson coefficients are values at 1 of certain parabolic Kazhdan-Lusztig polynomials for affine symmetric groups. These q-analogues of Littlewood-Richardson multiplicities coincide with those previously introduced in terms of ribbon tableaux.Comment: 51 pages, 10 Figures, Minor corrections and typos, and some new comments concerning Soergel's character formula for tilting module

Stabilized plethysms for the classical Lie groups

The plethysms of the Weyl characters associated to a classical Lie group by the symmetric functions stabilize in large rank. In the case of a power sum plethysm, we prove that the coefficients of the decomposition of this stabilized form on the basis of Weyl characters are branching coefficients which can be determined by a simple algorithm. This generalizes in particular some classical results by Littlewood on the power sum plethysms of Schur functions. We also establish explicit formulas for the outer multiplicities appearing in the decomposition of the tensor square of any irreducible finite dimensional module into its symmetric and antisymmetric parts. These multiplicities can notably be expressed in terms of the Littlewood-Richardson coefficients

Factorization of the canonical bases for higher level Fock spaces

The level l Fock space admits canonical bases G_e and G_\infty. They correspond to U_{v}(hat{sl}_{e}) and U_{v}(sl_{\infty})-module structures. We establish that the transition matrices relating these two bases are unitriangular with coefficients in N[v]. Restriction to the highest weight modules generated by the empty l-partition then gives a natural quantization of a theorem by Geck and Rouquier on the factorization of decomposition matrices which are associated to Ariki-Koike algebras.Comment: The last version generalizes and proves the main conjecture of the previous one. Final versio

kk-Schur functions and affine Schubert calculus

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website: http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates and corrections since v1. This material is based upon work supported by the National Science Foundation under Grant No. DMS-065264

Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals

Field of study: Mathematics.Dr. Calin Chindris, Dissertation Supervisor.Includes vita."May 2018."This thesis is devoted to the combinatorial and geometric study of certain multiplicities, which we call generalized Littlewood-Richardson coefficients. These are sums of products of single Littlewood-Richardson coefficients, and the specific ones we study describe the branching rules for the direct sum and diagonal embeddings of GL(n) as well as the decompositions of extremal weight crystals of type A+. By representing these multiplicities as dimensions of weight spaces of quiver semi-invariants, we use quiver theory to prove their saturation and describe necessary and sufficient conditions for them to be nonzero, culminating in statements similar to Horn's classical conjecture. We then use these conditions to prove various combinatorial properties, including how these multiplicities can be factored and that these numbers in certain cases satisfy the same conjectures as single Littlewood-Richardson coefficients. Finally, we provide a polytopal description of these multiplicities and prove that their positivity can be computed in strongly polynomial time.Includes bibliographical references (pages 120-124)