Graded Specht Modules as Bernstein–Zelevinsky Derivatives of the RSK Model

Abstract

We clarify the links between the graded Specht construction of modules over cyclotomic Hecke algebras and the Robinson-Schensted-Knuth (RSK) construction for quiver Hecke algebras of type $A$, which was recently imported from the setting of representations of $p$-adic groups. For that goal we develop a theory of crystal derivative operators on quiver Hecke algebra modules that categorifies the Berenstein–Zelevinsky strings framework on quantum groups and generalizes a graded variant of the classical Bernstein–Zelevinsky derivatives from the $p$-adic setting. Graded cyclotomic decomposition numbers are shown to be a special subfamily of the wider concept of RSK decomposition numbers.

Simple modules for quiver Hecke algebras and the Robinson–Schensted–Knuth correspondence

“…Here 𝐿 Remark 2.3. (See also [17,Remark 2.4]) The Kleshchev-Ram construction of a proper standard module out of a given multisegment 𝔪 ∈ 𝔐 is less canonical than how it may appear in our current presentation. More precisely, the construction depends on a choice of a total order on the set  (that is, on ℤ).…”

“…For $$L= L_\mathfrak {m}\in \operatorname{Irr}_{\mathcal {D}}$$, we also write $$\mathfrak {b}(L)=\mathfrak {b}(\mathfrak {m})$$, $$\mathfrak {e}(L)=\mathfrak {e}(\mathfrak {m})$$ and $$\mathrm{wt}(L) = \mathrm{wt}(\mathfrak {m})$$. Remark (See also [17, Remark 2.4]) The Kleshchev–Ram construction of a proper standard module out of a given multisegment $$\mathfrak {m}\in \mathfrak {M}$$ is less canonical than how it may appear in our current presentation. More precisely, the construction depends on a choice of a total order on the set $$\mathcal {I}$$ (that is, on $$\mathbb {Z}$$).…”

“…In a subsequent work of the author [17], it is shown that the RSK construction developed here may be viewed as a generalization of the Specht construction for cyclotomic Hecke algebras [12], and its graded variant [9].…”

“…𝐻) = Λ(𝐿 1 , 𝐿 2 ) + ⋯ + Λ(𝐿 1 , 𝐿 𝑘 ) holds, for 𝐻 ∈ Irr  , such that 𝐻⟨ℎ⟩ is the simple head of 𝐿 2 • ⋯ •𝐿 𝑘 .Proof. The equivalence of (1) and (2) is [21, Lemma 2.7], together with the observation that Λ(𝐿 1 , 𝐻) = Λ(𝐿 1 , 𝐻⟨ℎ⟩).Conditions (2) and (3) are equivalent because of the relation(17), the fact thatwt(𝐻) = wt(𝐿 2 • ⋯ •𝐿 𝑘 ) = wt(𝐿 2 ) + ⋯ + wt(𝐿 𝑘 ) ,and linearity of the form ( , ) on 𝑄. □ For a normal sequence (𝐿 1 , … , 𝐿 𝑘 ), let 𝐻⟨ℎ⟩ be the simple head of 𝐿1• ⋯ •𝐿 𝑘 with 𝐻 ∈ Irr  .…”