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Coxeter quotient at the Hecke specialization q=1

Statement

Let $n \geq 2$ , $V$ be a complex vector space, $U_{i} \in {End}_{C} (V)$ , $δ \in C$ , and $q \in C^{\times}$ satisfy $δ = q + q^{- 1}$ and

$U_{i}^{2} = δ U_{i}, U_{i} U_{i + 1} U_{i} = U_{i}, U_{i + 1} U_{i} U_{i + 1} = U_{i + 1}, U_{i} U_{j} = U_{j} U_{i} (∣ i - j ∣ > 1) .$

Put $B_{i} = q I_{V} - U_{i}$ , and let $ρ_{n} : B_{n} \to GL (V)$ be the homomorphism $ρ_{n} (σ_{i}) = B_{i}$ from the preceding Temperley–Lieb–Hecke construction. At $q = 1$ one has $δ = 2$ and $B_{i} = I_{V} - U_{i}$ . These operators satisfy $B_{i}^{2} = I_{V}$ , so $ρ_{n}$ factors through the canonical quotient $π_{n} : B_{n} \to S_{n}$ as a representation $S_{n} \to GL (V)$ . The $q = 1$ Hecke generator relations are then the type-A Coxeter relations. For general $q$ , the roots of the quadratic relation are $q$ and $- q^{- 1}$ ; they coincide when $q^{2} = - 1$ , and no diagonalizability is asserted. Involutivity is not imposed by the Hecke relation and need not hold.

Statement references

Temperley–Lieb projectors give a Hecke braid representation

The generic type-A Hecke algebra

The symmetric group has the Coxeter presentation

The braid group surjects onto the symmetric group

Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group

Facts & Assumptions

Given: The operator family, parameter relation, and braid-group representation from Temperley–Lieb projectors give a Hecke braid representation.

[F1]

Temperley–Lieb projectors give a Hecke braid representation: the $B_{i}$ satisfy the adjacent braid and distant-commutation relations and $(B_{i} - q I_{V}) (B_{i} + q^{- 1} I_{V}) = 0$ .

[F2]

The generic type-A Hecke algebra: the specialized generators have quadratic relation $(T_{i} - v) (T_{i} + 1) = 0$ , together with type-A braid and distant-commutation relations.

[F3]

The symmetric group has the Coxeter presentation: the adjacent transpositions $s_{i}$ present $S_{n}$ with relators $s_{i}^{2} = 1$ , the adjacent braid relations, and distant commutation.

[F4]

The braid group surjects onto the symmetric group: the canonical quotient sends $σ_{i} \mapsto s_{i}$ (in the library's transported indexing convention).

[F5]

Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group: a generator assignment satisfying all presentation relators extends uniquely to a homomorphism.

Proof

1.1F1givenalgebra

At $q = 1$ , the parameter relation gives $δ = 2$ . For every $i$ , $B_{i}^{2} = (I_{V} - U_{i})^{2} = I_{V} - 2 U_{i} + U_{i}^{2} = I_{V}$ , using $U_{i}^{2} = 2 U_{i}$ from [F1].

1.2F1algebra

The general quadratic in [F1] factors as $(B_{i} - q I_{V}) (B_{i} + q^{- 1} I_{V}) = 0$ , whose scalar roots are $q$ and $- q^{- 1}$ ; equality of those roots is equivalent to $q^{2} = - 1$ . This polynomial relation alone asserts no diagonalization.

2.1F1F3F5step 1.1construct

By [F1], the $B_{i}$ also satisfy the adjacent braid and distant-commutation relators. Along with step 1.1, these are exactly the Coxeter relators in [F3], so [F5] gives a homomorphism ${\tilde{ρ}}_{n} : S_{n} \to GL (V)$ with ${\tilde{ρ}}_{n} (s_{i}) = B_{i}$ .

3.1F1F4step 2.1algebra

The composite ${\tilde{ρ}}_{n} \circ π_{n}$ , with $π_{n}$ from [F4], and the braid representation $ρ_{n}$ from [F1] agree on every generator $σ_{i}$ , since both send it to $B_{i}$ . The generators determine a homomorphism from $B_{n}$ , so $ρ_{n} = {\tilde{ρ}}_{n} \circ π_{n}$ .

3.2F1F2F3step 1.1step 2.1algebra

The Hecke representation of the preceding theorem sends $T_{i}$ to $q B_{i}$ , which equals $B_{i}$ at $q = 1$ . The generic relation $(T_{i} - v) (T_{i} + 1) = 0$ in [F2] then becomes $T_{i}^{2} = I_{V}$ ; together with the braid and distant relations these are exactly the Coxeter generator relations in [F3]. Thus the specialized Hecke generators act by the same generator assignment as ${\tilde{ρ}}_{n}$ .

4.1F1algebra∎

To see that involutivity is not forced for general $q$ , take $V = C$ , $U_{i} = 0$ for every $i$ , $q = 2$ , and $δ = 5 / 2$ . All Temperley–Lieb relations hold, while $B_{i} = 2 I_{V}$ and $B_{i}^{2} = 4 I_{V} \neq I_{V}$ .

Remarks

At $q = 1$ the Coxeter quotient is obtained by adding involutions to the braid presentation. At general $q$ , the representation remains a braid/Hecke representation; no symmetric-group action follows unless the involution relations are separately verified.

Coxeter quotient at the Hecke specialization q=1

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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