[Paper Review] A KLR Grading of the Brauer Algebras

Li Ge|arXiv (Cornell University)|Sep 3, 2014

Algebraic structures and combinatorial models10 references7 citations

TL;DR

This paper constructs a Z-graded algebra $G_n(\delta)$ over a field of characteristic 0 with KLR-like relations, proving it is isomorphic to the Brauer algebra $B_n(\delta)$. The isomorphism establishes a non-trivial Z-grading on $B_n(\delta)$ and provides an explicit homogeneous cellular basis, demonstrating that $B_n(\delta)$ is a graded cellular algebra via a KLR-style presentation.

ABSTRACT

We construct a naturally $\mathbb Z$-graded algebra $\mathscr G_n(δ)$ over $R$ with KLR-like relations and give an explicit isomorphism between $\mathscr G_n(δ)$ and $\mathscr B_n(δ)$, the Brauer algebras over $R$, when $R$ is a field of characteristic 0. This isomorphism allows us to exhibit a non-trivial $\mathbb Z$-grading on the Brauer algebras over a field of characteristic 0. As a byproduct of the proof, we also construct an explicit homogeneous cellular basis for $\mathscr G_n(δ)$.

Motivation & Objective

• To construct a Z-graded algebra $G_n(\delta)$ over a field of characteristic 0 with relations analogous to those of cyclotomic quiver Hecke algebras.
• To establish an isomorphism between $G_n(\delta)$ and the Brauer algebra $B_n(\delta)$, thereby endowing $B_n(\delta)$ with a non-trivial Z-grading.
• To construct an explicit homogeneous cellular basis for $G_n(\delta)$, proving that $B_n(\delta)$ is a graded cellular algebra.

Proposed method

• Define $G_n(\delta)$ via generators $e(i), y_k, \psi_k, \epsilon_k$ and relations analogous to KLR algebras of type A.
• Construct a set of homogeneous elements $\psi^{st}$ indexed by up-down tableaux, with degree $\deg \psi^{st} = \deg s + \deg t$, forming a candidate basis.
• Prove that $G_n(\delta)$ is spanned by $\{ \psi^{st} \}$ using a cellular-like multiplication rule (Proposition 5.27), showing $\dim G_n(\delta) \leq (2n-1)!$.
• Establish a surjective algebra homomorphism $G_n(\delta) \to B_n(\delta)$ by mapping generators to standard generators of $B_n(\delta)$, using seminormal forms and known relations.
• Prove injectivity by showing the basis $\{ \psi^{st} \}$ has $ (2n-1)!! $ elements, matching $\dim B_n(\delta)$, hence the map is an isomorphism.
• Verify that the basis $\{ \psi^{st} \}$ satisfies the axioms of a graded cellular basis, including homogeneity and $*$-involution.

Experimental results

Research questions

• RQ1 Can the Brauer algebra $B_n(\delta)$ over a field of characteristic 0 be endowed with a non-trivial Z-grading?
• RQ2 Is there a KLR-like presentation for the Brauer algebra that realizes such a grading?
• RQ3 Does the Brauer algebra admit a homogeneous cellular basis analogous to that of cyclotomic quiver Hecke algebras?
• RQ4 Can the isomorphism between $G_n(\delta)$ and $B_n(\delta)$ be established via a cellular basis construction?

Key findings

• The algebra $G_n(\delta)$ is isomorphic to the Brauer algebra $B_n(\delta)$ over a field of characteristic 0, as proven by Theorem B.
• The isomorphism implies that $B_n(\delta)$ inherits a non-trivial Z-grading from $G_n(\delta)$, making it a graded algebra.
• The set $\{ \psi^{st} \}$ forms a homogeneous cellular basis for $G_n(\delta)$, and hence for $B_n(\delta)$, as shown in Theorem C.
• The dimension of $G_n(\delta)$ is bounded above by $ (2n-1)!! $, matching $\dim B_n(\delta)$, which confirms the basis spans and is linearly independent.
• The basis $\{ \psi^{st} \}$ satisfies a cellular multiplication rule (Proposition 5.27), confirming its cellular structure.
• The quotient $G_n(\delta)/E_n(\delta)$, where $E_n(\delta)$ is the two-sided ideal generated by $\epsilon_k$, is isomorphic to the cyclotomic quiver Hecke algebra $R^\Lambda_n$, as shown in Theorem 7.79.