243 research outputs found
Applications of the Brauer complex: card shuffling, permutation statistics, and dynamical systems
By algebraic group theory, there is a map from the semisimple conjugacy
classes of a finite group of Lie type to the conjugacy classes of the Weyl
group. Picking a semisimple class uniformly at random yields a probability
measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is
proved that this measure agrees with a second measure on conjugacy classes of
the Weyl group induced by a construction of Cellini using the affine Weyl
group. Formulas for Cellini's measure in type are found. This leads to new
models of card shuffling and has interesting combinatorial and number theoretic
consequences. An analysis of type C gives another solution to a problem of
Rogers in dynamical systems: the enumeration of unimodal permutations by cycle
structure. The proof uses the factorization theory of palindromic polynomials
over finite fields. Contact is made with symmetric function theory.Comment: One change: we fix a typo in definition of f(m,k,i,d) on page 1
Affine and toric hyperplane arrangements
We extend the Billera-Ehrenborg-Readdy map between the intersection lattice
and face lattice of a central hyperplane arrangement to affine and toric
hyperplane arrangements. For arrangements on the torus, we also generalize
Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure
The Poincar\'e-extended ab-index
Motivated by a conjecture concerning Igusa local zeta functions for
intersection posets of hyperplane arrangements, we introduce and study the
Poincar\'e-extended ab-index, which generalizes both the ab-index and the
Poincar\'e polynomial. For posets admitting R-labelings, we give a
combinatorial description of the coefficients of the extended ab-index, proving
their nonnegativity. In the case of intersection posets of hyperplane
arrangements, we prove the above conjecture of the second author and Voll as
well as another conjecture of the second author and K\"uhne. We also define the
pullback ab-index generalizing the cd-index of face posets for oriented
matroids. Our results recover, generalize and unify results from
Billera-Ehrenborg-Readdy, Bergeron-Mykytiuk-Sottile-van Willigenburg,
Saliola-Thomas, and Ehrenborg. This connection allows us to translate our
results into the language of quasisymmetric functions, and-in the special case
of symmetric functions-make a conjecture about Schur positivity.Comment: Expanded implications for matroids (Remark 2.15, Examples 2.16-17),
connections to Zeta functions (Remark 2.26), and a new section about
quasisymmetric functions (Section 3
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