80 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
Completeness of the Bethe Ansatz for an Open q -Boson System with Integrable Boundary Interactions
We employ a discrete integral-reflection representation of the double affine Hecke algebra of type C∨C at the critical level q = 1 , to endow the open finite q-boson system with integrable boundary interactions at the lattice ends. It is shown that the Bethe Ansatz entails a complete basis of eigenfunctions for the commuting quantum integrals in terms of Macdonald’s three-parameter hyperoctahedral Hall–Littlewood polynomials.Fil: van Diejen, Jan Felipe. Universidad de Talca; ChileFil: Emsiz, Erdal. Pontificia Universidad Católica de Chile; ChileFil: Zurrián, Ignacio Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin
A nonsymmetric version of Okounkov's BC-type interpolation Macdonald polynomials
Symmetric and nonsymmetric interpolation Laurent polynomials are introduced
with the interpolation points depending on and a -tuple of parameters
. For the principal specialization
the symmetric interpolation Laurent polynomials reduce to
Okounkov's -type interpolation Macdonald polynomials and the nonsymmetric
interpolation Laurent polynomials become their nonsymmetric variants. We expand
the symmetric interpolation Laurent polynomials in the nonsymmetric ones. We
show that Okounkov's -type interpolation Macdonald polynomials can also be
obtained from their nonsymmetric versions using a one-parameter family of
actions of the finite Hecke algebra of type in terms of Demazure-Lusztig
operators. In the Appendix we give some experimental results and conjectures
about extra vanishing.Comment: 30 pages, 9 figures; v4: experimental results and conjectures added
about extra vanishin
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