Abstract
In this extended abstract we present colored generalizations of the symmetric algebra and its Koszul dual, the exterior algebra. The symmetric group Sn acts on the multilinear components of these algebras. While Sn acts trivially on the multilinear components of the colored symmetric algebra, we use poset topology techniques to describe the representation on its Koszul dual. We introduce an Sn-poset of weighted subsets that we call the weighted boolean algebra and we prove that the multilinear components of the colored exterior algebra are Sn- isomorphic to the top cohomology modules of its maximal intervals. We show that the two colored Koszul dual algebras are Koszul in the sense of Priddy et al.
Highlights
Let k denote a field of characteristic not equal to 2 and V be a finite dimensional k-vector space
The tensor algebra T (V ) = n≥0 V ⊗n is the free associative algebra generated by V, where V ⊗n denotes the tensor product of n copies of V and V ⊗0 := k
The Koszul dual A! of A is the algebra A! := A(V ∗, R⊥) = T (V ∗)/ R⊥. It follows from relations (1.1) and (1.2) that Λ(V ∗) is the Koszul dual associative algebra to S(V )
Summary
Let k denote a field of characteristic not equal to 2 and V be a finite dimensional k-vector space. The tensor algebra T (V ) = n≥0 V ⊗n is the free associative algebra generated by V , where V ⊗n denotes the tensor product of n copies of V and V ⊗0 := k. Let V ∗ := Hom(V, k) denote the vector space dual to V. Recall that for an associative algebra A = A(V, R) := T (V )/ R generated on a finite dimensional vector space V and (quadratic) relations R ⊆ V ⊗2 there is another algebra A! It follows from relations (1.1) and (1.2) that Λ(V ∗) is the Koszul dual associative algebra to S(V ). S(n) ∼=Sn 1n and Λ(n) ∼=Sn sgnn, where 1n and sgnn are respectively the trivial and the sign representations of Sn
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