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The two-tableau pairing at a singular parameter

Example

For $λ = (2, 1)$ put $x = 1 + T_{1}$ and $b_{0} = x$ , $b_{1} = x T_{2}$ , $b_{2} = x T_{2} T_{1}$ in the row permutation module. Its form is $diag (1, q, q^{2})$ and its generator action matrices, with coordinate row vectors, are $R_{1} = (\begin{matrix} q & 0 & 0 \\ 0 & 0 & 1 \\ 0 & q & q - 1 \end{matrix})$ , $R_{2} = (\begin{matrix} 0 & 1 & 0 \\ q & q - 1 & 0 \\ 0 & 0 & q \end{matrix})$ . The ordinary polytabloid basis is $p_{0} = z = x T_{2} (1 - q^{- 1} T_{1}) = b_{1} - q^{- 1} b_{2}$ and $p_{1} = z T_{2} = q b_{0} + (q - 1) b_{1} - b_{2}$ . Their embedding has unit determinant on its $b_{0}, b_{2}$ coordinates, and their form is $B = (\begin{array}{cc} q + 1 & q^{2} \\ q^{2} & q^{3} + q \end{array}) .$ In the Murphy quotient let $c_{0} = x$ , $c_{1} = x T_{2}$ modulo the higher shape $(3)$ . Its form and generator action matrices, with coordinate row vectors, are $G = (\begin{array}{cc} q + 1 & - 1 \\ - 1 & q^{2} + 1 \end{array}), M_{1} = (\begin{array}{cc} q & 0 \\ - 1 & - 1 \end{array}), M_{2} = (\begin{array}{cc} 0 & 1 \\ q & q - 1 \end{array}) .$ On the ordinary basis the matrices are $P_{1} = (\begin{matrix} - 1 & 0 \\ - q^{2} & q \end{matrix})$ and $P_{2} = M_{2}$ . The map $Φ (p_{0}) = c_{0}$ , $Φ (p_{1}) = (q - 1) c_{0} - c_{1}$ has matrix $A = (\begin{matrix} 1 & 0 \\ q - 1 & - 1 \end{matrix})$ ; it satisfies $P_{i} A = A ((q - 1) I - M_{i})$ and $A G A^{T} = B$ . The same-shape perfect cross-pairing on the bases $(c_{0}, c_{1})$ and $(p_{0}, p_{1})$ has matrix $Q = (\begin{matrix} 0 & q \\ q & q (q - 1) \end{matrix})$ with determinant $- q^{2}$ , a unit; its entries are the row-form coefficients before quotienting. Thus the exact self-form multiplier is $1$ . Both Gram determinants equal $q (1 + q + q^{2})$ . At $q = - 1$ over characteristic zero these forms are nonsingular with determinant $- 1$ , even though $[2]_{q} = 0$ ; at $q = 1$ in characteristic $2$ both reduce to $(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$ and have radical $0$ . This distinguishes a singular algebra parameter from degeneracy of this particular module form. No choice principle is used.

Facts & Assumptions

Given: The Hecke algebra $H = H_{R, q} (S_{3})$ over a commutative ring $R$ with $q \in R^{\times}$ , the shape $λ = (2, 1) = λ^{'}$ , the row module $M (λ)$ with $x = x_{λ} = 1 + T_{1}$ and the distinguished basis ${b_{0}, b_{1}, b_{2}}$ , the polytabloid module $P (λ) = z_{λ} H$ with $z_{λ} = x T_{2} (1 - q^{- 1} T_{1})$ , the permutation form $β$ and the Murphy cell $C_{M} (λ)$ with classes $c_{0}, c_{1}$ and form $G$ , together with the comparison theorem and the perfect cross-pairing of Ordinary Specht modules are the signed conjugate Murphy cells and Murphy cells are the same-shape contragredient Specht duals.

[F1]

$H$ has generators $T_{1}, T_{2}$ with $(T_{i} - q) (T_{i} + 1) = 0$ , braid relation $T_{1} T_{2} T_{1} = T_{2} T_{1} T_{2}$ and standard basis ${T_{w}}$ ; $x T_{1} = q x$ (The generic type-A Hecke algebra, The row permutation module: distinguished basis and Hecke action).

[F2]

The distinguished basis ${b_{0}, b_{1}, b_{2}} = {x, x T_{s_{2}}, x T_{s_{2} s_{1}}}$ of $M (λ)$ satisfies $β (b_{i}, b_{j}) = δ_{i j} q^{i}$ and is the row basis of $D_{λ} = {1, s_{2}, s_{2} s_{1}}$ (Hecke row permutation modules and ordinary Dipper-James polytabloids, The row permutation module: distinguished basis and Hecke action).

[F3]

$C_{M} (λ) = M (λ) / (M (λ) \cap I^{⊳ λ})$ has basis $c_{t} = x T_{d (t)}$ for the standard tableaux $t$ , and its form satisfies $m_{u s} m_{t v} \equiv G (c_{s}, c_{t}) m_{u v} (m o d I^{⊳ λ})$ ; the ideal $I^{⊳ λ}$ contains $x_{(3)} = b_{0} + b_{1} + b_{2}$ (The Murphy cellular basis, the cell law and the Murphy form).

[F4]

$Φ ⁣ : P (λ) \to C_{M} (λ)^{θ}$ is an isomorphism with $Φ (z) = c_{0}$ and unit $u_{λ} (q) = q^{ℓ (w_{0} (S_{λ^{'}})) - ℓ (w_{λ})} = q^{1 - 1} = 1$ for $λ = (2, 1)$ ; the pairing $B_{λ} (m_{s}, p) = β (x T_{d (s)}, p)$ is perfect with the explicit entries $B_{λ} (c_{s}, e_{t}) = q^{ℓ (d (s))} a_{d (s), t}$ (Ordinary Specht modules are the signed conjugate Murphy cells, Murphy cells are the same-shape contragredient Specht duals).

Verification

technique · direct

1.1F1F2algebra

Row module. From [F1], $b_{0} T_{1} = x T_{1} = q x = q b_{0}$ , $b_{1} T_{1} = x T_{2} T_{1} = b_{2}$ and $b_{2} T_{1} = x T_{2} T_{1}^{2} = x T_{2} ((q - 1) T_{1} + q) = (q - 1) b_{2} + q b_{1}$ ; also $b_{0} T_{2} = b_{1}$ , $b_{1} T_{2} = x T_{2}^{2} = x ((q - 1) T_{2} + q) = (q - 1) b_{1} + q b_{0}$ and $b_{2} T_{2} = x T_{2} T_{1} T_{2} = x T_{1} T_{2} T_{1} = q x T_{2} T_{1} = q b_{2}$ , using the braid relation. In the basis $(b_{0}, b_{1}, b_{2})$ with coordinate row vectors these are exactly the matrices $R_{1}$ and $R_{2}$ of the statement, and $β (b_{i}, b_{i}) = q^{i}$ by [F2] because $ℓ (1) = 0$ , $ℓ (s_{2}) = 1$ , $ℓ (s_{2} s_{1}) = 2$ .

2.1step 1.1F2F4algebra

Polytabloids and their form. Since $z_{λ} = x T_{2} (1 - q^{- 1} T_{1})$ by [F4] and $y_{λ^{'}} = 1 - q^{- 1} T_{1}$ , the displayed expressions $p_{0} = b_{1} - q^{- 1} b_{2}$ and $p_{1} = z T_{2} = b_{1} T_{2} - q^{- 1} b_{2} T_{2} = q b_{0} + (q - 1) b_{1} - b_{2}$ follow from step 1.1. Their coordinates on $(b_{0}, b_{2})$ give the matrix $(\begin{matrix} 0 & - q^{- 1} \\ q & - 1 \end{matrix})$ with determinant $q \cdot q^{- 1} = 1$ , a unit over every $R$ ; hence the two polytabloids span a free rank-two split submodule. The row action gives $p_{0} T_{1} = - p_{0}$ , $p_{1} T_{1} = - q^{2} p_{0} + q p_{1}$ , $p_{0} T_{2} = p_{1}$ and $p_{1} T_{2} = q p_{0} + (q - 1) p_{1}$ . Thus their span is stable, has precisely the displayed matrices $P_{1}, P_{2}$ , and equals $z H$ because it contains $z = p_{0}$ and $z T_{2} = p_{1}$ . Weighted inner products against the diagonal form of [F2] give $B = (\begin{matrix} q + 1 & q^{2} \\ q^{2} & q^{3} + q \end{matrix})$ : explicitly $B_{00} = q + q^{- 2} \cdot q^{2} = q + 1$ , $B_{01} = q (q - 1) + (- q^{- 1}) (- 1) q^{2} = q (q - 1) + q = q^{2}$ , and $B_{11} = q^{2} + (q - 1)^{2} q + q^{2} = q^{3} + q$ . Its determinant is $q (q^{2} + q + 1)$ .

2.2step 1.1F3algebra

The Murphy quotient. In $C_{M} (λ)$ the higher vector $x_{(3)} = b_{0} + b_{1} + b_{2}$ vanishes by [F3], so $b_{2} = - c_{0} - c_{1}$ and $c_{1} = x T_{2} = b_{1}$ . Then $c_{0} T_{1} = q x = q c_{0}$ and $c_{1} T_{1} = b_{2} = - c_{0} - c_{1}$ , giving $M_{1}$ ; also $c_{0} T_{2} = c_{1}$ and $c_{1} T_{2} = (q - 1) b_{1} + q b_{0} = (q - 1) c_{1} + q c_{0}$ , giving $M_{2}$ . The defining products of [F3] give $x^{2} = (q + 1) x$ and $x T_{2} x = b_{1} + b_{2} = x_{(3)} - x \equiv - x$ . Also $x T_{2}^{2} x = (q - 1) x T_{2} x + q x^{2} \equiv (q^{2} + 1) x$ . Reading the coefficient of $m_{t_{0} t_{0}} = x$ therefore gives $G (c_{0}, c_{0}) = q + 1$ , $G (c_{0}, c_{1}) = - 1$ and $G (c_{1}, c_{1}) = q^{2} + 1$ ; equivalently $G = (\begin{matrix} q + 1 & - 1 \\ - 1 & q^{2} + 1 \end{matrix})$ with determinant $q^{3} + q^{2} + q$ .

3.1step 2.1step 2.2F4algebra∎

Comparison, cross-pairing and specializations. The comparison theorem of [F4] gives $Φ (z) = c_{0}$ , hence $Φ (p_{1}) = Φ (z T_{2}) = c_{0} θ (T_{2}) = (q - 1) c_{0} - c_{1}$ and $A = (\begin{matrix} 1 & 0 \\ q - 1 & - 1 \end{matrix})$ ; direct matrix multiplication verifies $P_{i} A = A ((q - 1) I - M_{i})$ for $i = 1, 2$ and $A G A^{T} = B$ , the latter being the form identity with the unit $u_{λ} = 1$ . The cross-pairing entries $B_{λ} (c_{s}, p_{t}) = β (b_{d (s)}, p_{t})$ on the standard bases give $Q = (\begin{matrix} 0 & q \\ q & q (q - 1) \end{matrix})$ with $\det Q = - q^{2} \in A^{\times}$ , so the pairing is perfect as in [F4]. The two Gram determinants agree with $q (1 + q + q^{2})$ . At $q = - 1$ in characteristic zero, $B = (\begin{matrix} 0 & 1 \\ 1 & - 2 \end{matrix})$ and $G = (\begin{matrix} 0 & - 1 \\ - 1 & 2 \end{matrix})$ have determinant $- 1$ , although $[2]_{q} = 1 + q = 0$ ; at $q = 1$ in characteristic $2$ , both reduce to $(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$ with determinant $1$ , hence with zero radical. All identities are Laurent-polynomial identities, so they specialize to every $R$ ; no choice principle is used.

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Sources

Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, author-hosted prepublication manuscript, 79 PDF pages (standard reference, not scraped)
G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer 1978; author-hosted scan (standard reference, not scraped)
Steen Ryom-Hansen, On the denominators of Young's seminormal basis, arXiv:0904.4243v5, 28 PDF pages (standard reference, not scraped)