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Motivated by work of Coxeter (1957), we study a class of algebras associated to Coxeter groups, which we term generalized nil-Coxeter algebras. We construct the first finite-dimensional examples other than usual nil-Coxeter algebras; these form a $2$-parameter type $A$ family that we term $NC_A(n,d)$. We explore the combinatorial properties of these algebras, including the Coxeter word basis, length function, maximal words, and their connection to Khovanovs categorification of the Weyl algebra. Our broader motivation arises from complex reflection groups and the Broue-Malle-Rouquier freeness conjecture (1998). With generic Hecke algebras over real and complex groups in mind, we show that the first finite-dimensional examples $NC_A(n,d)$ are in fact the only ones, outside of the usual nil-Coxeter algebras. The proofs use a diagrammatic calculus akin to crystal theory.
We investigate a certain linear combination $K(vec{x})=K(a;b,c,d;e,f,g)$ of two Saalschutzian hypergeometric series of type ${_4}F_3(1)$. We first show that $K(a;b,c,d;e,f,g)$ is invariant under the action of a certain matrix group $G_K$, isomorphic
We develop tools for classification of contraction algebras and apply these to solve the problem on classification up to isomorphism of 8 and 9 dimensional algebras corresponding to 3-fold flops. We prove that there is only one up to isomorphism cont
Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Grobner basis theory and gener
We exhibit a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several authors. A conseq
In 2010, Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoids form a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type $A$). In this paper, we give a family