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The ordered-basis symmetry gives the normalized Hecke tensor action

Statement

Let $F = Q (q)$ , $N \geq 1$ , and $n \geq 0$ . Let $H_{n} (F, q)$ be the type-A Hecke algebra generated by $t_{1}, \dots, t_{n - 1}$ with braid relations and $(t_{i} - q) (t_{i} + q^{- 1}) = 0$ ; put $H_{0} = H_{1} = F$ . On $V_{N}^{\otimes n}$ define $T_{i}$ by applying the operator of The ordered-basis Hecke operator and its normalization in positions $i, i + 1$ . There is a unital algebra homomorphism $ρ_{n} : H_{n} (F, q) \to {End}_{F} (V_{N}^{\otimes n})$ with $ρ_{n} (t_{i}) = T_{i}$ . Its matrices are regular at $q = 1$ , where $T_{i}$ is the transposition of tensor positions $i, i + 1$ . This is an action into an endomorphism algebra; it makes no assertion that $ρ_{n}$ is injective.

Facts & Assumptions

Given: $F = Q (q)$ , $N \geq 1$ , $n \geq 0$ , and the normalized type-A Hecke presentation.

[F1]

The presentation of $H_{n}$ is the specialization of the universal Coxeter Hecke presentation in Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy with type $A_{n - 1}$ and common parameter $q$ .

[F2]

The universal presentation has the usual quotient universal property; The standard basis of the generic Hecke algebra and base change proves its standard basis and its base-change compatibility.

[F3]

The operator of The ordered-basis operator is a Hecke symmetry satisfies the quadratic and braid relations, and at $q = 1$ it is the flip.

Proof

Given: The data in [F1]–[F3].

1.1givenF1F3algebra

For $n \geq 2$ , each $T_{i}$ satisfies $(T_{i} - q I) (T_{i} + q^{- 1} I) = 0$ by [F3]. Adjacent operators satisfy the braid relation because, on the three positions they meet, this is exactly the identity $T_{1} T_{2} T_{1} = T_{2} T_{1} T_{2}$ proved in [F3]; tensoring with identities on the other positions preserves that equality. If $∣ i - j ∣ > 1$ , then $T_{i}$ and $T_{j}$ act on disjoint tensor factors, so on every elementary tensor each changes only its own pair and the two orders of application give the same tensor and coefficient. Thus the family $(T_{i})$ satisfies every defining relation in [F1].

2.1step 1.1F2F3∎

By the quotient universal property in [F2], the assignment $t_{i} \mapsto T_{i}$ extends uniquely to a unital $F$ -algebra homomorphism $ρ_{n}$ . For $n = 0, 1$ the algebra and endomorphism image are the scalar algebra and the unique unital map supplies the same conclusion. Each matrix entry of every $T_{i}$ is a Laurent polynomial in $q$ , so it is regular at $q = 1$ ; [F3] identifies the specialized operators as adjacent flips. The standard-basis theorem in [F2] concerns the abstract algebra and does not identify it with its image under $ρ_{n}$ .

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Sources

Meinolf Geck, Modular Representations of Hecke Algebras (standard reference, not scraped)
Alexei Davydov and Alexander Molev, A categorical approach to classical and quantum Schur–Weyl duality (standard reference, not scraped)