Galleries and q-analogs in combinatorial representation theory

Schur functions and their q-analogs constitute an interesting branch of combinatorial representation theory. For Schur functions one knows several combinatorial formulas regarding their expansion in terms of monomial symmetric functions, their structure constants and their branching coefficients. In this thesis we prove q-analogs of these formulas for Hall-Littlewood polynomials. We give combinatorial formulas for the expansion of Hall-Littlewood polynomials in terms of monomial symmetric functions, for their structure constants and their branching coefficients. Specializing these formulas we get new proofs for the formulas involving Schur functions. As a combinatorial tool we use the gallery model introduced by Gaussent and Littelmann and show its relation to the affine Hecke algebra. All assertions are then proven in the more general context of the Macdonald basis of the spherical Hecke algebra. We show a commutation formula in the affine Hecke algebra with which we obtain a Demazure character formula involving galleries. We give a geometric interpretation of Kostka numbers and Demazure multiplicities of a complex reductive algebraic group using the affine Grassmanian of its Langlands dual group. As a further application we prove some first results regarding the positivity of Kostka-Foulkes coefficients

Galleries, Hall-Littlewood polynomials and structure constants of the spherical Hecke algebra

We provide a combinatorial description of the coefficients appearing in the expansion of Hall-Littlewood polynomials in terms of monomial symmetric functions. We also give a Littlewood-Richardson rule for Hall-Littlewood polynomials. For proving this we use galleries to calculate Satake coefficients and structure constants of spherical Hecke algebras with arbitrary parameters.Comment: 28 pages, v3: revised version, to appear in IMR

Shadows in Coxeter groups

For a given $w$ in a Coxeter group $W$ the elements $u$ smaller than $w$ in Bruhat order can be seen as the end-alcoves of stammering galleries of type $w$ in the Coxeter complex $\Sigma$. We generalize this notion and consider sets of end-alcoves of galleries that are positively folded with respect to certain orientation $\phi$ of $\Sigma$. We call these sets shadows. Positively folded galleries are closely related to the geometric study of affine Deligne-Lusztig varieties, MV polytopes, Hall-Littlewood polynomials and many more agebraic structures. In this paper we will introduce various notions of orientations and hence shadows and study some of their algorithmic properties.Comment: 30 pages, 8 figures, revised and final versio