434 research outputs found
Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux
We introduce a new family of noncommutative analogues of the Hall-Littlewood
symmetric functions. Our construction relies upon Tevlin's bases and simple
q-deformations of the classical combinatorial Hopf algebras. We connect our new
Hall-Littlewood functions to permutation tableaux, and also give an exact
formula for the q-enumeration of permutation tableaux of a fixed shape. This
gives an explicit formula for: the steady state probability of each state in
the partially asymmetric exclusion process (PASEP); the polynomial enumerating
permutations with a fixed set of weak excedances according to crossings; the
polynomial enumerating permutations with a fixed set of descent bottoms
according to occurrences of the generalized pattern 2-31.Comment: 37 pages, 4 figures, new references adde
One-skeleton galleries, the path model and a generalization of Macdonald's formula for Hall-Littlewood polynomials
We give a direct geometric interpretation of the path model using galleries
in the skeleton of the Bruhat-Tits building associated to a semi-simple
algebraic group. This interpretation allows us to compute the coefficients of
the expansion of the Hall-Littlewood polynomials in the monomial basis. The
formula we obtain is a "geometric compression" of the one proved by Schwer, its
specialization to the case turns out to be equivalent to
Macdonald's formula.Comment: 43 pages, 3 pictures, some improvements in the presentation,
semistandard tableaux for type B and C define
Plethysm and lattice point counting
We apply lattice point counting methods to compute the multiplicities in the
plethysm of . Our approach gives insight into the asymptotic growth of
the plethysm and makes the problem amenable to computer algebra. We prove an
old conjecture of Howe on the leading term of plethysm. For any partition
of 3,4, or 5 we obtain an explicit formula in and for the
multiplicity of in .Comment: 25 pages including appendix, 1 figure, computational results and code
available at http://thomas-kahle.de/plethysm.html, v2: various improvements,
v3: final version appeared in JFoC
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