1,156 research outputs found
Tableaux in the Whitney Module of a Matroid
The Whitney module of a matroid is a natural analogue of the tensor algebra
of the exterior algebra of a vector space that takes into account the
dependencies of a matroid. In this paper we indicate the role that tableaux can
play in describing the Whitney module. We will use our results to describe a
basis of the Whitney module of a certain class of matroids known as freedom
matroids (also known as Schubert, or shifted matroids). The doubly multilinear
submodule of the Whitney module is a representation of the symmetric group. We
will describe a formula for the multiplicity of hook shapes in this
representation in terms of no broken circuit sets
Products of Linear Forms and Tutte Polynomials
Let \Delta be a finite sequence of n vectors from a vector space over any
field. We consider the subspace of \operatorname{Sym}(V) spanned by \prod_{v
\in S} v, where S is a subsequence of \Delta. A result of Orlik and Terao
provides a doubly indexed direct sum of this space. The main theorem is that
the resulting Hilbert series is the Tutte polynomial evaluation
T(\Delta;1+x,y). Results of Ardila and Postnikov, Orlik and Terao, Terao, and
Wagner are obtained as corollaries.Comment: Minor changes. Accepted for publication in European Journal of
Combinatoric
A Short Proof of Gamas's Theorem
If \chi^\lambda is the irreducible character of the symmetric group S_n
corresponding to the partition \lambda of n then we may symmetrize a tensor v_1
\otimes ... \otimes v_n by \chi^\lambda. Gamas's theorem states that the result
is not zero if and only if we can partition the set {v_i} into linearly
independent sets whose sizes are the parts of the transpose of \lambda. We give
a short and self-contained proof of this fact
Equality of symmetrized tensors and the coordinate ring of the flag variety
In this note we give a transparent proof of a result of da Cruz and Dias da
Silva on the equality of symmetrized decomposable tensors. This will be done by
explaining that their result follows from the fact that the coordinate ring of
a flag variety is a unique factorization domain.Comment: 5 page
A type B analog of the Lie representation
International audienceWe describe a type B analog of the much studied Lie representation of the symmetric group. The nth Lie representation of Sn restricts to the regular representation of Sn−1, and our generalization mimics this property. Specifically, we construct a representation of the type B Weyl group Bn that restricts to the regular representation of Bn−1. We view both of these representations as coming from the internal zonotopal algebra of the Gale dual of the corresponding reflection arrangements
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