[Paper Review] Affine cellularity of affine Yokonuma-Hecke algebras

Weideng Cui|arXiv (Cornell University)|Oct 9, 2015

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper establishes an explicit algebra isomorphism between the affine Yokonuma-Hecke algebra $ω_{r,n}(q)$ and a direct sum of matrix algebras over tensor products of affine Hecke algebras of type $A$, proving that $ω_{r,n}(q)$ is affine cellular. As a key result, it shows the algebra has finite global dimension when $q$ is not a root of the Poincaré polynomial, and recovers its modular representation theory.

ABSTRACT

We establish an explicit algebra isomorphism between the affine Yokonuma-Hecke algebra $\widehat{Y}_{r,n}(q)$ and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type $A.$ As an application of this result, we show that $\widehat{Y}_{r,n}(q)$ is affine cellular in the sense of Koenig and Xi, and further prove that it has finite global dimension when the parameter $q$ is not a root of the Poincaré polynomial. As another application, we also recover the modular representation theory of $\widehat{Y}_{r,n}(q)$ previously obtained in [CW].

Motivation & Objective

• To establish an explicit algebra isomorphism between the affine Yokonuma-Hecke algebra $ω_{r,n}(q)$ and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type $A$.
• To prove that $ω_{r,n}(q)$ is affine cellular in the sense of Koenig and Xi.
• To show that $ω_{r,n}(q)$ has finite global dimension when $q$ is not a root of the Poincaré polynomial.
• To recover the modular representation theory of $ω_{r,n}(q)$ previously obtained in [CW].

Proposed method

• Construct an auxiliary algebra $ω E_{r,n}$ as a direct sum of matrix algebras with coefficients in tensor products of extended affine Hecke algebras of type $A$, using Lusztig's framework.
• Prove that $ω E_{r,n}$ satisfies the axiomatic properties $P_1, P_2, P_3, P_4$ from [C2] for affine cellular algebras.
• Establish an explicit algebra isomorphism $ω Y_{r,n}(q) o ω E_{r,n}$ via a map $ψ$ defined on generators and extended linearly.
• Use the isomorphism to transfer properties from $ω E_{r,n}$ to $ω Y_{r,n}(q)$, including affine cellularity and finite global dimension.
• Apply results from [KX2] and [C2] on affine cellular algebras to deduce structural properties such as stratification of derived module categories.
• Leverage the isomorphism to recover the modular representation theory of $ω Y_{r,n}(q)$ over fields of characteristic $p$ not dividing $r$.

Experimental results

Research questions

• RQ1 Is the affine Yokonuma-Hecke algebra $ω_{r,n}(q)$ affine cellular in the sense of Koenig and Xi?
• RQ2 Can an explicit algebra isomorphism be constructed between $ω_{r,n}(q)$ and a direct sum of matrix algebras over tensor products of affine Hecke algebras of type $A$?
• RQ3 Under what conditions on $q$ does $ω_{r,n}(q)$ have finite global dimension?
• RQ4 Does the isomorphism allow for a recovery of the modular representation theory of $ω_{r,n}(q)$ previously established in [CW]?
• RQ5 What is the structure of the derived module category of $ω_{r,n}(q)$, and how does it relate to the asymptotic algebra?

Key findings

• An explicit algebra isomorphism is constructed between $ω_{r,n}(q)$ and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type $A$, proving that $ω_{r,n}(q)$ is affine cellular.
• The algebra $ω_{r,n}(q)$ has finite global dimension when $q$ is not a root of the Poincaré polynomial.
• The parameter set of simple $ω_{r,n}(q)$-modules over a field of characteristic zero containing a primitive $r$-th root of unity is a finite union of affine spaces.
• The derived module category of $ω_{r,n}(q)$ admits a stratification whose strata are the derived module categories of affine $k$-algebras $B_l$, as a consequence of the affine cellularity and idempotent generators of layers.
• The modular representation theory of $ω_{r,n}(q)$ over an algebraically closed field of characteristic $p$ not dividing $r$ is recovered via the isomorphism.
• When $q$ is not a root of the Poincaré polynomial, $ω_{r,n}(q)$ is affine quasi-hereditary.