Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures

Abstract

The underlying motivation is to apply the theory of noncommutative Gröbner bases in free associative algebras to the construction of universal associative envelopes for nonassociative structures defined by multilinear operations. Trilinear operations were classified by the author and Peresi in 2007. In her Ph.D. thesis of 2012, Elgendy studied the universal associative envelopes of nonassociative triple systems obtained by applying these trilinear operations to the 2-dimensional simple associative triple system. In these notes I use computer algebra to extend some aspects of her work to the 4-dimensional and 6-dimensional simple associative triple systems.

Lefschetz theory for exterior algebras and fermionic diagonal coinvariants

“…Given a term order <, for any nonzero element f ∈ ∧{Θ n }, let lm(f ) be the largest monomial θ S under the total order < such that θ S appears with nonzero coefficient in f . If I ⊆ ∧{Θ n } is a two-sided ideal, let The set N (I) of monomials descends to a C-basis of the quotient ∧{Θ n }/I; this is the standard monomial basis with respect to < (see for example [4]).…”

“…Rosas proves [17, Proof of Thm. 13 (4)] that the multiplicity of the Schur function s For any 0 ≤ k ≤ n − 1 and all i + j < n, we have…”