81 research outputs found
Odd length for even hyperoctahedral groups and signed generating functions
We define a new statistic on the even hyperoctahedral groups which is a
natural analogue of the odd length statistic recently defined and studied on
Coxeter groups of types and . We compute the signed (by length)
generating function of this statistic over the whole group and over its maximal
and some other quotients and show that it always factors nicely. We also
present some conjectures
Peak algebras, paths in the Bruhat graph and Kazhdan-Lusztig polynomials
We give a new characterization of the peak subalgebra of the algebra of
quasisymmetric functions and use this to construct a new basis for this
subalgebra. As an application of these results we obtain a combinatorial
formula for the Kazhdan-Lusztig polynomials which holds in complete generality
and is simpler and more explicit than any existing one. We then show that, in a
certain sense, this formula cannot be simplified.Comment: 31 page
Alternating subgroups of Coxeter groups
We study combinatorial properties of the alternating subgroup of a Coxeter
group, using a presentation of it due to Bourbaki.Comment: 39 pages, 3 figure
Mixed Bruhat operators and Yang-Baxter equations for Weyl groups
We introduce and study a family of operators which act in the span of a Weyl
group and provide a multi-parameter solution to the quantum Yang-Baxter
equations of the corresponding type. Our operators generalize the "quantum
Bruhat operators" that appear in the explicit description of the multiplicative
structure of the (small) quantum cohomology ring of .
The main combinatorial applications concern the "tilted Bruhat order," a
graded poset whose unique minimal element is an arbitrarily chosen element
. (The ordinary Bruhat order corresponds to the case .) Using the
mixed Bruhat operators, we prove that these posets are lexicographically
shellable, and every interval in a tilted Bruhat order is Eulerian. This
generalizes well known results of Verma, Bjorner, Wachs, and Dyer.Comment: 19 page
A Construction of Coxeter Group Representations (II)
An axiomatic approach to the representation theory of Coxeter groups and
their Hecke algebras was presented in [1]. Combinatorial aspects of this
construction are studied in this paper. In particular, the symmetric group case
is investigated in detail. The resulting representations are completely
classified and include the irreducible ones.Comment: 24 pages, shorter background; to appear in J. Algebr
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