50 research outputs found
On a curious variant of the -module
We introduce a variant of the much-studied representation of the
symmetric group , which we denote by Our variant gives rise
to a decomposition of the regular representation as a sum of {exterior} powers
of modules This is in contrast to the theorems of
Poincar\'e-Birkhoff-Witt and Thrall which decompose the regular representation
into a sum of symmetrised modules. We show that nearly every known
property of has a counterpart for the module suggesting
connections to the cohomology of configuration spaces via the character
formulas of Sundaram and Welker, to the Eulerian idempotents of Gerstenhaber
and Schack, and to the Hodge decomposition of the complex of injective words
arising from Hochschild homology, due to Hanlon and Hersh.Comment: 26 pages, 2 tables. To appear in Algebraic Combinatorics. Parts of
this paper are included in arXiv:1803.0936
An elementary approach to the quasipolynomiality of the Kronecker coefficients
The Kronecker coefficients are the structure constants for the restriction of
irreducible representations of the general linear group GL(nm) into
irreducibles for the subgroup GL(n)xGL(m). In this work we study the piecewise
quasipolynomial nature of the Kronecker function using tools from polyhedral
geometry. By bounding the lengths of the partitions, we write the Kronecker
function in terms of coefficients of vector partition functions. We illustrate
the power of this approach in the first nontrivial case: we give exact formulas
and an upper bound for the Kronecker coefficients, and derive other properties.
An additional advantage of this approach is that asymptotic estimates for
dilations are computable using techniques of analytic combinatorics in several
variables. We also show how our approach can be applied to other families of
structure constants.Comment: 44 pages; some results on chambers added, revised introductio
On the topology of two partition posets with forbidden block sizes
AbstractWe study two subposets of the partition lattice obtained by restricting block sizes. The first consists of set partitions of {1,…,n} with block size at most k, for k≤n−2. We show that the order complex has the homotopy type of a wedge of spheres, in the cases 2k+2≥n and n=3k+2. For 2k+2>n, the posets in fact have the same Sn−1-homotopy type as the order complex of Πn−1, and the Sn-homology representation is the “tree representation” of Robinson and Whitehouse. We present similar results for the subposet of Πn in which a unique block size k≥3 is forbidden. For 2k≥n, the order complex has the homotopy type of a wedge of (n−4)-spheres. The homology representation of Sn can be simply described in terms of the Whitehouse lifting of the homology representation of Πn−1
On counting permutations by pairs of congruence classes of major index
For a fixed positive integer n, let S_n denote the symmetric group of n!
permutations on n symbols, and let maj(sigma) denote the major index of a
permutation sigma. For positive integers k<m not greater than n and
non-negative integers i and j, we give enumerative formulas for the cardinality
of the set of permutations sigma in S_n with maj(sigma) congruent to i mod k
and maj(sigma^(-1)) congruent to j mod m. When m divides n-1 and k divides n,
we show that for all i,j, this cardinality equals (n!)/(km).Comment: 8 page
The reflection representation in the homology of subword order
We investigate the homology representation of the symmetric group on
rank-selected subposets of subword order.
We show that the homology module for words of bounded length, over an
alphabet of size
decomposes into a sum of tensor powers of the -irreducible
indexed by the partition recovering, as a special case, a theorem of
Bj\"orner and Stanley for words of length at most For arbitrary ranks we
show that the homology is an integer combination of positive tensor powers of
the reflection representation , and conjecture that this
combination is nonnegative. We uncover a curious duality in homology in the
case when one rank is deleted.
We prove that the action on the rank-selected chains of subword order is a
nonnegative integer combination of tensor powers of , and show
that its Frobenius characteristic is -positive and supported on the set
Our most definitive result describes the Frobenius characteristic of the
homology for an arbitrary set of ranks, plus or minus one copy of the Schur
function as an integer combination of the set
We conjecture that this
combination is nonnegative, establishing this fact for particular cases.Comment: 30 pages; typos corrected; sections have been reorganised. To appear
in Algebraic Combinatoric