58 research outputs found
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
Combinatorial and Algebraic Structure in Orlik–Solomon Algebras
AbstractThe Orlik–Solomon algebra A(G) of a matroid G is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex hyperplane arrangement realizing G. On the other hand, some features of the matroid G are reflected in the algebraic structure of A(G). In this mostly expository article, we describe recent developments in the construction of algebraic invariants of A(G). We develop a categorical framework for the statement and proof of recently discovered isomorphism theorems which suggests a possible setting for classification theorems. Several specific open problems are formulated
Resonant bands, Aomoto complex, and real 4-nets
The resonant band is a useful notion for the computation of the nontrivial
monodromy eigenspaces of the Milnor fiber of a real line arrangement. In this
article, we develop the resonant band description for the cohomology of the
Aomoto complex. As an application, we prove that real 4-nets do not exist.Comment: 23 pages, 7 figure
On Some Quadratic Algebras I : Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials
We study some combinatorial and algebraic properties of certain quadratic
algebras related with dynamical classical and classical Yang-Baxter equations.
One can find more details about the content of present paper in Extended
Abstract.Comment: Dedicated to the memory of Alain Lascoux (1944-2013). Preprint
RIMS-1817, 172 page
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