We study the crystal structure on categories of graded modules over algebras which categorify the negative half of the quantum Kac-Moody algebra associated to a symmetrizable Cartan data. We identify this crystal with Kashiwara's crystal for the corresponding negative half of the quantum Kac-Moody algebra. As a consequence, we show the simple graded modules for certain cyclotomic quotients carry the structure of highest weight crystals, and hence compute the rank of the corresponding Grothendieck group.Comment: 56 pages, 6 eps files. v2 corrects typo

We give a short exposition of new and known results on the "standard method" of identifying a hidden subgroup of a nonabelian group using a quantum computer. AbstractWe give a short exposition of new and known results on the "standard method"of identifying a hidden subgroup of a nonabelian group using a quantum computer.

The permutation representation afforded by a Coxeter group W acting on the cosets of a standard parabolic subgroup inherits many nice properties from W such as a shellable Bruhat order and a flat deformation over Z[q] to a representation of the corresponding Hecke algebra. In this paper we define a larger class of "quasiparabolic" subgroups (more generally, quasiparabolic W -sets), and show that they also inherit these properties. Our motivating example is the action of the symmetric group on fixed-point-free involutions by conjugation.

We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund's bijection ζ exchanging the pairs of statistics (area, dinv) and (bounce, area) on Dyck paths, and the Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions. We also relate our combinatorial constructions to representation theory. We derive new formulas for the Poincaré polynomials of certain affine Springer fibers and describe a connection to the theory of finite dimensional representations of DAHA and nonsymmetric Macdonald polynomials. Proposition 1.1. If m and n are coprime then m-stable affine permutations label the alcoves in a certain simplex D m n which is isometric to the m-dilated fundamental alcove. In particular, the number of m-stable affine permutations equals m n−1 .The simplex D m n (first defined in [10,27]) plays the central role in our study. We show that the alcoves in it naturally label various algebraic and geometric objects such as cells in certain affine Springer fibres and nonsymmetric Macdonald polynomials at q m = t n . We provide a clear combinatorial dictionary that allows one to pass from one description to another.We define two maps A, PS between the m-stable affine permutations and m n-parking functions and prove the following results about them. Theorem 1.2. Maps A and PS satisfy the following properties:(1) The map A is a bijection for all m and n.(2) The map PS is a bijection for m = kn ± 1. For m = kn + 1, it is equivalent to the Pak-Stanley labeling of Shi regions. 1 2 EUGENE GORSKY, MIKHAIL MAZIN, AND MONICA VAZIRANI (3) The map PS ○ A −1 generalizes the bijection ζ constructed by Haglund in [18]. More concretely, if one takes m = n + 1 and restricts the maps A and PS to minimal length right coset representatives of S n S n , then PS ○ A −1 specializes to Haglund's ζ. Remark 1.3. For m = n + 1 the bijection A is similar to the Athanasiadis-Linusson [5] labeling of Shi regions, but actually differs from it. Conjecture 1.4. The map PS is bijective for all relatively prime m and n.The map PS has an important geometric meaning. In [24] Lusztig and Smelt considered a certain Springer fibre F m n in the affine flag variety and proved that it can be paved by m n−1 affine cells. In [14,15] a related subvariety of the affine Grassmannian has been studied under the name of Jacobi factor, and a bijection between its cells and the Dyck paths in m × n rectangle has been constructed. In [20] Hikita generalized this combinatorial analysis and constructed a bijection between the cells in the affine Springer fiber and m n-parking functions (in slightly different terminology). He gave a quite involved combinatorial formula for the dimension of ...

When one expands a Schur function in terms of the irreducible characters of the symplectic (or orthogonal) group, the coefficient of the trivial character is 0 unless the indexing partition has an appropriate form. A number of q, t-analogues of this fact were conjectured in [10]; the present paper proves most of those conjectures, as well as some new identities suggested by the proof technique. The proof involves showing that a nonsymmetric version of the relevant integral is annihilated by a suitable ideal of the affine Hecke algebra, and that any such annihilated functional satisfies the desired vanishing property. This does not, however, give rise to vanishing identities for the standard nonsymmetric Macdonald and Koornwinder polynomials; we discuss the required modification to these polynomials to support such results.