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A place permutation is not the nondegenerate Hecke generator at xi=-1

Statement

Let $M = C$ be the trivial ${g l}_{2} (C)$ -module and let $V = C^{2}$ . On $M \otimes V^{\otimes 2} ≅ V \otimes V$ , let $P = P_{12}$ interchange the two natural factors. On the basis vector $v = e_{1} \otimes e_{1}$ , one has $P v = v$ . For the nondegenerate type-A Hecke algebra at parameter $ξ = - 1$ , its generator $T_{1}$ must satisfy $T_{1}^{2} = - 2 T_{1} - 1.$ Consequently the assignment $T_{1} \mapsto P$ is not a representation of this Hecke algebra. This refutes the naive identification of the degenerate place-permutation generator with the nondegenerate generator at this parameter; it does not rule out other Hecke–KLR or Schur–Weyl maps.

Facts & Assumptions

Given: $F = C$ , $ξ = - 1$ , the trivial ${g l}_{2}$ -module $M = C$ , $V = C^{2}$ with basis $e_{1}, e_{2}$ , and the place permutation $P$ on $M \otimes V^{\otimes 2} ≅ V \otimes V$ .

[F1]

Category O, KLR, and Hecke route conventions: the Category-O tensor-functor generator belongs to the degenerate affine route, while $ξ$ denotes the separate nondegenerate finite-Hecke parameter.

[F2]

The nondegenerate root-of-unity level-one Hecke regime: the nondegenerate branch has $ξ \neq 1$ and finite quantum characteristic.

[F3]

Affine and cyclotomic type-A Hecke algebras: the nondegenerate quadratic relation is $T_{i}^{2} = (q - 1) T_{i} + q$ .

[F4]

Arakawa-Suzuki Casimir operators on tensor functors: $s_{i} = P_{i, i + 1}$ is the place-permutation operator in the degenerate tensor action.

[F5]

The Casimir and place-permutation operators satisfy the degenerate affine Hecke relations: the operators $s_{i} = P_{i, i + 1}$ satisfy the degenerate affine Hecke relations on $M \otimes V^{\otimes d}$ .

[F6]

The elementary tensors of two bases form the product basis of the tensor product: tensor products of basis vectors form a basis of the tensor product, so $e_{1} \otimes e_{1} \neq 0$ .

Proof

1.1givenF4F6algebra

The place permutation satisfies $P (e_{1} \otimes e_{1}) = e_{1} \otimes e_{1} = v$ by [F4], and $v \neq 0$ because tensor products of basis vectors form a basis [F6]; hence $P^{2} v = v$ .

1.2givenF2F3F4F6algebra

The parameter $ξ = - 1 \neq 1$ has order two in $C$ , so [F2] puts it in the nondegenerate finite- $e$ branch with $e = 2$ . On the nonzero vector $v = e_{1} \otimes e_{1}$ , [F4] gives $P v = v$ and [F6] gives $v \neq 0$ . The quadratic relation [F3] at $q = ξ = - 1$ is $T_{1}^{2} = - 2 T_{1} - 1$ ; assigning $T_{1} = P$ would require $v = - 2 v - v = - 3 v$ , hence $4 v = 0$ , a contradiction over $C$ . Thus the defining relation fails.

2.1givenF1F5algebra∎

This only refutes the specified assignment: other parameter conversions or homomorphisms remain outside this calculation, and the cited degenerate action [F5] is still valid.

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Sources

Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, AMS University Lecture Series 15 manuscript, complete 79-page book text (standard reference, not scraped)
Tomoyuki Arakawa and Takeshi Suzuki, Lie algebras and degenerate affine Hecke algebras of type A, J. Algebra 209 (1998), complete 16-page preprint (standard reference, not scraped)