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Rank-two type-A cell modules at a singular parameter

Statement

Let $F$ be a field and specialize the level-one, rank-two type-A Hecke algebra to $H = F [T] / ((T - Q) (T + 1)),$ where $Q \in F^{\times}$ . Its row Murphy elements are $m_{(2)} = 1 + T$ and $m_{(1, 1)} = 1$ , ordered by $(2) ▹ (1, 1)$ . They form a cellular basis, and the cell modules have forms $Q + 1$ and $1$ . At $Q = - 1$ , $m_{(2)}^{2} = 0$ , its cell form vanishes, and the $(1, 1)$ cell form remains nonzero.

The row-Murphy cell module convention is distinct from the existing ordinary Specht convention. The exact comparison is Ordinary Specht modules are the signed conjugate Murphy cells: the ordinary Specht label $λ$ corresponds to the sign twist of the Murphy cell at the conjugate partition $λ^{'}$ . This example does not identify same-label row Murphy cells with those Specht modules.

Facts & Assumptions

Given: A field $F$ , $Q \in F^{\times}$ , the rank-two type-A Hecke algebra, and the row-Murphy cell convention.

[F1]

The multiplicative cyclotomic Hecke algebra: at level one the finite generators obey $(T - Q) (T + 1) = 0$ .

[F2]

Cyclotomic Murphy elements: at level one $u_{λ} = 1$ ; $x_{(2)} = 1 + T$ and $x_{(1, 1)} = 1$ .

[F3]

Cell data, multipartition dominance, and higher-cell ideals: a cellular basis uses an involution-swapped basis and the fixed-first-index multiplication law modulo higher cells.

[F4]

Cell modules, their forms, and radicals: the cell form is the same-cell multiplication coefficient modulo higher cells.

[F5]

Ordinary Specht modules are the signed conjugate Murphy cells: the ordinary Specht module at $λ$ is the sign twist of the Murphy cell at $λ^{'}$ ; there is no untwisted same-label identification.

Proof

Given: $F$ , $Q$ , $T$ , and the candidate Murphy elements $1 + T$ and $1$ .

1.1F1F2construct

The quadratic relation is $T^{2} = (Q - 1) T + Q$ . The row subgroup of $(2)$ is $S_{2}$ , giving row sum $1 + T$ ; the row subgroup of $(1, 1)$ is trivial, giving $1$ . The partition dominance inequalities give $(2) ▹ (1, 1)$ .

1.2F1F2F3

Monic division by $(T - Q) (T + 1)$ gives basis $1, T$ for $H$ . Since $T = (1 + T) - 1$ , the pair $1 + T, 1$ is also a basis. Direct multiplication gives $(1 + T)^{2} = 1 + 2 T + T^{2} = (Q + 1) (1 + T) .$ Also $T (1 + T) = Q (1 + T)$ , while $T = (1 + T) - 1 \equiv - 1$ modulo the upper cell. Thus each cell has the required fixed-row action, and the involution fixes both basis elements; this verifies the cellular basis directly.

2.1F1F2F4step 1.2algebra

By [F4] the upper cell form is the coefficient $Q + 1$ in the displayed square, and the lower cell form is $1 \cdot 1 = 1$ . At $Q = - 1$ , $(T + 1)^{2} = 0$ and $T + 1$ is nonzero because it is a basis element. The upper form vanishes and the lower form remains $1$ . The algebra is $F [ϵ] / (ϵ^{2})$ with $ϵ = T + 1$ . For any simple right module $S$ , $ϵ S$ is a submodule; if it were nonzero then $ϵ S = S$ , forcing $ϵ^{2} S = S$ , contrary to $ϵ^{2} = 0$ . Hence $ϵ S = 0$ , and $S$ is the unique one-dimensional module with $T = - 1$ .

3.1F5step 1.1∎

The cross-link in [F5] places the ordinary Specht module at the conjugate Murphy label and applies the sign twist. The calculation above concerns the row-Murphy labels $(2)$ and $(1, 1)$ , so the comparison requires both operations and does not identify the row cell at $λ$ with an untwisted Specht module at $λ$ .

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Sources

Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete author-uploaded preliminary lecture manuscript (1998) (standard reference, not scraped)
Richard Dipper, Gordon James, and Andrew Mathas, Cyclotomic q-Schur algebras, Mathematische Zeitschrift 229 (1998), 385–416 (standard reference, not scraped)