A diamond lemma for Hecke-type algebras

Abstract

In this paper we give a version of Bergman's diamond lemma which applies to certain monoidal categories presented by generators and relations. In particular, it applies to: the Coxeter presentation of the symmetric groups, the quiver Hecke algebras of Khovanov-Lauda-Rouquier, the Webster tensor product algebras, and various generalizations of these.We also give an extension of Manin-Schechtmann theory to non-reduced expressions.

Algebraic Weaves and Braid Varieties

“…If we forget the decoration, this poset becomes a poset on the set of words with Demazure product ∆. Its analogue for all expressions of ∆ and covering relations ss → s replaced by ss → e was defined by Elias [29] as an extension of the second higher Bruhat order to necessarily reduced words. It was used in the proof of the main result of the work [29], which we translated to our language as Theorem 4.6.…”

“…Its analogue for all expressions of ∆ and covering relations ss → s replaced by ss → e was defined by Elias [29] as an extension of the second higher Bruhat order to necessarily reduced words. It was used in the proof of the main result of the work [29], which we translated to our language as Theorem 4.6. Our weaves thus resemble saturated chains in the secong higher Bruhat order, which in turn can be seen as elements of the third higher Bruhat order.…”

“…Proof. The theorem follows from the main result of [29], which we briefly recall. For part (a), consider the graph where vertices correspond to braid words and edges to braid moves (that is, 6-or 4-valent vertices).…”

“…The mutation equivalence of any weaves yielding toric charts follows from the more general Theorem 4.6, which states that, under technical conditions that are satisfied in the weaves pertaining to Theorem 1.8, any two Demazure weaves between the same two braid words are related by a sequence of equivalence moves and mutations. Note that Theorem 4.6 is a translation of a result of B. Elias [29] to our weaves framework.…”

On Bergman's Diamond Lemma for Ring Theory

“…To get more recent trends for the Diamond Lemma, I would like to cite, among others, Chenavier [5], Chenavier and Lucas [6], Elias [7] and Tsuchioka [11]. There the reader will find much more information on the lemma and see some practices as an application to representation theory, as an example.…”