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The Murphy tableau elements form a cellular basis

Statement

Let $R$ be a commutative ring, $q \in R^{\times}$ , and $H = H_{R, q} (S_{n})$ (The generic type-A Hecke algebra). Let $Λ$ be the set of partitions of $n$ ordered by dominance $⊵$ (Dominance order on partitions), and for $λ \in Λ$ let $T (λ) = S t d (λ)$ be the set of standard $λ$ -tableaux (Tableaux and standard tableaux). Then the Murphy basis theorem holds: the family $M = {m_{s t} : λ ⊢ n, s, t \in S t d (λ)}$ is an $R$ -basis of $H$ , and $(M, Λ)$ is a cellular basis of $H$ in the sense of Cellular bases and cell modules. Explicitly: (i) $m_{s t}^{*} = m_{t s}$ for all $λ$ and $s, t$ ; and (ii) for every $λ$ , every $t \in S t d (λ)$ and every $h \in H$ there exist $r_{v} \in R$ with $m_{s t} h \equiv \sum_{v \in S t d (λ)} r_{v} m_{s v} (m o d H^{⊳ λ}),$ where $H^{⊳ λ}$ is spanned by the $m_{u v}$ with $μ ⊳ λ$ . The theorem holds over every commutative coefficient ring with $q \in R^{\times}$ and assumes no semisimplicity; the total number of Murphy elements is $\sum_{λ ⊢ n} ∣ S t d (λ) ∣^{2} = n!$ (The sum of squares of the standard tableau numbers), matching the rank of $H$ .

Facts & Assumptions

Given: A commutative ring $R$ , a unit $q$ , the type-A Hecke algebra and the Murphy elements $m_{s t} = T_{d (s)}^{*} x_{λ} T_{d (t)}$ in the Statement.

[F1]

The independent integral construction proves that these exact elements form a base-change-compatible basis; their dominance spans are two-sided ideals, star interchanges the indices, and their multiplication satisfies the fixed-first-index cell law over every commutative coefficient ring with unit parameter (The Murphy cellular basis, the cell law and the Murphy form).

[F2]

The elements $x_{λ}$ and $d (t)$ are the row symmetrizer and distinguished tableau permutations of Murphy elements and Murphy tableau elements, using the standard Hecke basis of The standard basis of the generic type-A Hecke algebra. The dominance poset, tableau sets and cardinality sum are Dominance order on partitions, Tableaux and standard tableaux and The sum of squares of the standard tableau numbers.

Proof

technique · direct

1.1givenF1F2algebra

Identify the objects in [F1]. Its row symmetrizer is $x_{λ} = \sum_{w \in S_{λ}} T_{w}$ , its permutation is the unique distinguished $d (t)$ carrying the row-filled tableau to $t$ , and its element is $T_{d (s)}^{*} x_{λ} T_{d (t)}$ . These are precisely [F2]'s Murphy elements. The proved integral basis in [F1] therefore gives the required $R$ -basis and $m_{s t}^{*} = m_{t s}$ , including the empty tableau when $n = 0$ .

2.1F1F2step 1.1algebra∎

Its dominance ideal $I^{⊳ λ}$ and the present $H^{⊳ λ}$ are the spans of exactly the same basis elements, so they are equal. The independent cell law says $m_{s t} h \equiv \sum_{v} r_{v} m_{s v}$ modulo this ideal, with coefficients independent of $s$ . Together with step 1.1 and the same finite dominance poset, these are exactly the cellular axioms and all asserted formulas. The basis has $\sum_{λ} ∣ Std (λ) ∣^{2} = n!$ elements by [F2]. Every part holds after every base change by [F1], so no specialized semisimplicity or factor inversion is needed. No choice principle is used.

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Sources

Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete prepublication manuscript (1998; expanded published edition: AMS University Lecture Series 15, 1999); complete 79-page manuscript, SHA-256 8cfeb611a95a3ff25cb95ad22bfd3a68475818301c34b04d131a8480a1b997e0 (standard reference, not scraped)