Parabolic Kazhdan-Lusztig R-polynomials for tight quotients of the symmetric groups

We give explicit closed combinatorial formulas for the parabolic Kazhdan-Lusztig R-polynomials of the tight quotients of the symmetric groups. We give two formulations of our result, one in terms of permutations and one in terms of Motzkin paths. As an application of our results we obtain explicit closed combinatorial formulas for certain sums and alternating sums of ordinary Kazhdan-Lusztig R-polynomials

Quasisymmetric functions and Kazhdan-Lusztig polynomials

We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatorially defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality

Odd and even major indices and one-dimensional characters for classical Weyl groups

We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz's identity, of the Gessel-Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs

Odd length: odd diagrams and descent classes

We define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets of permutations and acyclic orientations of the Turán graph, that the dimension of the odd Schubert variety associated to a permutation is the odd length of the permutation, and give several necessary conditions for a subset of [ n ] × [ n ] to be the odd diagram of a permutation. We also study the sign-twisted generating function of the odd length over descent classes of the symmetric groups

Permutations, tensor products, and Cuntz algebra automorphisms

We study the reduced Weyl groups of the Cuntz algebras O-n from a combinatorial point of view. Their elements correspond bijectively to certain permutations of n(r) elements, which we call stable. We mostly focus on the case r = 2 and general n. A notion of rank is introduced, which is subadditive in a suitable sense. Being of rank 1 corresponds to solving an equation which is reminiscent of the Yang-Baxter equation. Symmetries of stable permutations are also investigated, along with an immersion procedure that allows to obtain stable permutations of (n + 1)(2) objects starting from stable permutations of n(2) objects. A complete description of stable transpositions and of stable 3-cycles of rank 1 is obtained, leading to closed formulas for their number. Other enumerative results are also presented which yield lower and upper bounds for the number of stable permutations

Kazhdan-Lusztig polynomials, tight quotients and Dyck superpartitions

We give an explicit combinatorial product formula for the parabolic Kazhdan–Lusztig polynomials of the tight quotients of the symmetric group. This formula shows that these polynomials are always either zero or a monic power of q and implies the main result in [F. Brenti, Kazhdan–Lusztig and R-polynomials, Youngʼs lattice, and Dyck partitions, Pacific J. Math. 207 (2002) 257–286] on the parabolic Kazhdan–Lusztig polynomials of the maximal quotients. The formula depends on a new class of superpartitions

Comparing and characterizing some constructions of canonical bases from Coxeter systems

The Iwahori-Hecke algebra $\mathcal{H}$ of a Coxeter system $(W,S)$ has a "standard basis" indexed by the elements of $W$ and a "bar involution" given by a certain antilinear map. Together, these form an example of what Webster calls a pre-canonical structure, relative to which the well-known Kazhdan-Lusztig basis of $\mathcal{H}$ is a canonical basis. Lusztig and Vogan have defined a representation of a modified Iwahori-Hecke algebra on the free $\mathbb{Z}[v,v^{-1}]$-module generated by the set of twisted involutions in $W$, and shown that this module has a unique pre-canonical structure satisfying a certain compatibility condition, which admits its own canonical basis which can be viewed as a generalization of the Kazhdan-Lusztig basis. One can modify the parameters defining Lusztig and Vogan's module to obtain other pre-canonical structures, each of which admits a unique canonical basis indexed by twisted involutions. We classify all of the pre-canonical structures which arise in this fashion, and explain the relationships between their resulting canonical bases. While some of these canonical bases are related in a trivial fashion to Lusztig and Vogan's construction, others appear to have no simple relation to what has been previously studied. Along the way, we also clarify the differences between Webster's notion of a canonical basis and the related concepts of an IC basis and a $P$-kernel.Comment: 32 pages; v2: additional discussion of relationship between canonical bases, IC bases, and P-kernels; v3: minor revisions; v4: a few corrections and updated references, final versio

Degenerate flag varieties and the median Genocchi numbers

We study the \bG_a^M degenerations \Fl^a_\la of the type $A$ flag varieties \Fl_\la. We describe these degenerations explicitly as subvarieties in the products of Grassmanians. We construct cell decompositions of \Fl^a_\la and show that for complete flags the number of cells is equal to the normalized median Genocchi numbers $h_n$. This leads to a new combinatorial definition of the numbers $h_n$. We also compute the Poincar\' e polynomials of the complete degenerate flag varieties via a natural statistics on the set of Dellac's configurations, similar to the length statistics on the set of permutations. We thus obtain a natural $q$-version of the normalized median Genocchi numbers.Comment: 18 page

Classical symmetric functions in superspace

We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the classical bases, namely, the monomial symmetric functions, the elementary symmetric functions, the completely symmetric functions, and the power sums. Various basic results, such as the generating functions for the multiplicative bases, Cauchy formulas, involution operations as well as the combinatorial scalar product are also generalized.Comment: 21 pages, this supersedes the first part of math.CO/041230

Lattice Point Generating Functions and Symmetric Cones

We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically contrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out the specific cases of a symmetry group of type A (previously known) and types B and D (new). We obtain several applications of the special cases in type B, including identities involving permutation statistics and lecture hall partitions.Comment: 19 page