[Paper Review] The Calogero-Moser partition and Rouquier families for complex reflection groups

Maurizio Martino|ArXiv.org|Jan 10, 2008

Algebraic structures and combinatorial models21 references3 citations

TL;DR

This paper proposes a conjecture linking the Calogero-Moser partition of irreducible representations of complex reflection groups to Rouquier families in cyclotomic Hecke algebras. It proves the conjecture for wreath products $G(l,1,n) = \mathbb{Z}/l\mathbb{Z} \wr S_n$ by showing that Rouquier families refine the Calogero-Moser partition, with a geometric interpretation of block sizes and a connection to higher-level Fock spaces via residue conditions on power series.

ABSTRACT

Let $W$ be a complex reflection group. We formulate a conjecture relating blocks of the corresponding restricted rational Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. We verify the conjecture in the case that $W$ is a wreath product of a symmetric group with a cyclic group of order $l$.

Motivation & Objective

• To formulate a conjecture relating blocks of restricted rational Cherednik algebras (Calogero-Moser partition) to Rouquier families in cyclotomic Hecke algebras for complex reflection groups.
• To verify this conjecture in the case of wreath products $G(l,1,n) = \mathbb{Z}/l\mathbb{Z} \wr S_n$.
• To provide a geometric interpretation of the size of each Rouquier block in terms of residue conditions on power series.
• To connect the Calogero-Moser partition to higher-level Fock space representations of $U_v(\mathfrak{sl}_\infty)$ via combinatorial invariants.

Proposed method

• Define the restricted rational Cherednik algebra $\overline{H}_{\mathbf{k}}$ as a finite-dimensional quotient of the rational Cherednik algebra $H_{\mathbf{k}}$, with blocks defining the Calogero-Moser partition of $\mathrm{Irr}W$.
• Use the equivalence $H_{\mathbf{k}} \cong H_{\lambda\mathbf{k}}$ to normalize parameters and simplify the analysis of the Calogero-Moser partition.
• Establish a combinatorial correspondence between Rouquier families and the $J$-heart of multipartitions via residue conditions on power series $\mathrm{Res}^{\overline{\mathbf{m}}}_{\boldsymbol{\lambda}}(x^d)$.
• Prove that $\boldsymbol{\lambda} \sim_{R_{\mathbf{m}}} \boldsymbol{\mu}$ if and only if $\mathrm{Res}^{\overline{\mathbf{m}}}_{\boldsymbol{\lambda}}(x^d) = \mathrm{Res}^{\overline{\mathbf{m}}}_{\boldsymbol{\mu}}(x^d)$, using projections onto $x^k\mathbb{Z}[x^d,x^{-d}]$ components.
• Show that equal $J$-hearts imply $\boldsymbol{\lambda} \sim_{CM_{\mathbf{h}}} \boldsymbol{\mu}$ via Theorem 3.9, and use parameter scaling to relate $CM_{\mathbf{h}}$ and $CM_{d\mathbf{h}}$ partitions.
• Connect the Calogero-Moser partition to the weight of basis vectors $s_{\boldsymbol{\lambda}}$ in higher-level Fock space $F(\Lambda_{\mathbf{m}})$ of $U_v(\mathfrak{sl}_\infty)$.

Experimental results

Research questions

• RQ1 Does the partition of irreducible representations into Rouquier families refine the Calogero-Moser partition for complex reflection groups?
• RQ2 Can the size of each Rouquier block be geometrically interpreted via residue conditions on power series?
• RQ3 Is there a connection between the Calogero-Moser partition and the weight of vectors in higher-level Fock space representations?
• RQ4 Does the conjecture hold for wreath products $G(l,1,n)$, and if so, what is the underlying combinatorial mechanism?
• RQ5 How do the parameters $\mathbf{k}$ and $\mathbf{m}$ relate to the block structure in both the Cherednik and Hecke algebra settings?

Key findings

• The conjecture that Rouquier families refine the Calogero-Moser partition is verified for $G(l,1,n)$ via explicit combinatorial comparison of the two partitions.
• Two multipartitions $\boldsymbol{\lambda}, \boldsymbol{\mu} \in \mathcal{P}(l,n)$ lie in the same Rouquier family if and only if $\mathrm{Res}^{\overline{\mathbf{m}}}_{\boldsymbol{\lambda}}(x^d) = \mathrm{Res}^{\overline{\mathbf{m}}}_{\boldsymbol{\mu}}(x^d)$.
• The Calogero-Moser partition for $G(l,1,n)$ coincides with the partition into weights of basis vectors in the higher-level Fock space $F(\Lambda_{\mathbf{m}})$ of $U_v(\mathfrak{sl}_\infty)$.
• The $J$-heart of a multipartition determines its Calogero-Moser block, and equal $J$-hearts imply block equivalence under $CM_{\mathbf{h}}$.
• The parameter scaling property $\overline{H}_{\mathbf{k}} \cong \overline{H}_{d\mathbf{k}}$ implies that $\boldsymbol{\lambda} \sim_{CM_{\mathbf{h}}} \boldsymbol{\mu}$ if and only if $\boldsymbol{\lambda} \sim_{CM_{d\mathbf{h}}} \boldsymbol{\mu}$.
• The geometric size of each Rouquier block is determined by the number of multipartitions sharing the same residue condition $\mathrm{Res}^{\overline{\mathbf{m}}}_{\boldsymbol{\lambda}}(x^d)$.