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The finite Hecke algebra is non-canonically isomorphic to the group algebra of S_n

Statement

Assume the Axiom of Choice, used through Tits deformation. For every prime power $q$ there is an isomorphism of $C$ -algebras $H = e_{B} C [G] e_{B} ≅ C [S_{n}]$ , $G = {GL}_{n} (F_{q})$ , and no isomorphism sending every standard basis element $T_{w}$ to $w$ exists for $n \geq 2$ and $q \neq 1$ , since their quadratic relations differ: the deformation isomorphism is not canonical, does not identify the natural bases, and need not identify the simple modules of $H$ with those of $C [S_{n}]$ in any prescribed way. Consequently $H$ and $C [S_{n}]$ have the same number of simple modules and the same multiset of dimensions of simple modules, but no natural bijection of simple modules is asserted.

Facts & Assumptions

Given: A prime power $q$ , the group $G = {GL}_{n} (F_{q})$ with Borel $B$ , the finite Hecke algebra $H = e_{B} C [G] e_{B}$ with standard basis $T_{w}$ , the group algebra $C [S_{n}]$ with its basis $S_{n}$ , and the Axiom of Choice AC.

[F1]

AC holds, and Tits deformation gives $H_{q} (S_{n}) ≅ e_{B} C [{GL}_{n} (F_{q})] e_{B} ≅ C [S_{n}]$ , preserving the number and dimensions of simple modules; the isomorphism is produced by a formal-lifting and constructible-incidence argument (Tits deformation for the type-A Hecke algebra, The Axiom of Choice).

[F2]

The algebra $H_{q} (S_{n})$ in [F1] is the specialization at $v \mapsto q$ of the generic Hecke algebra $H_{v} (n)$ , and $C [S_{n}]$ is its specialization at $v \mapsto 1$ (Group algebra and finite-field specializations of the generic Hecke algebra).

[F3]

For every simple transposition $s_{i}$ one has $T_{s_{i}}^{2} = (q - 1) T_{s_{i}} + q 1_{H}$ in $H$ , with $1_{H} = T_{1} = e_{B}$ the unit (The type-A Iwahori-Hecke presentation of the finite Hecke algebra, The Bruhat double-coset basis of the finite Hecke algebra).

[F4]

The elements $T_{w}$ , $w \in S_{n}$ , form a $C$ -basis of $H$ , and in $C [S_{n}]$ the elements $w$ , $w \in S_{n}$ , form a basis; in particular $1$ and a simple transposition $s_{i}$ are linearly independent in $C [S_{n}]$ . [F3, given]

Proof

technique · direct

1.1F1F2

By [F2] the algebra $H_{q} (S_{n})$ of [F1] is $H$ and its specialization at $v \mapsto 1$ is $C [S_{n}]$ ; by [F1] there is an isomorphism $H ≅ C [S_{n}]$ of $C$ -algebras; any algebra isomorphism induces an equivalence between the categories of finite-dimensional modules, so it carries simple modules to simple modules and preserves their dimensions. Hence the number and the multiset of dimensions of simple modules agree.

1.2F3F4algebra

For $n \geq 2$ there is no unital algebra isomorphism $φ : H \to C [S_{n}]$ with $φ (T_{w}) = w$ for all $w$ . Indeed, applying such a $φ$ to the relation $T_{s_{i}}^{2} = (q - 1) T_{s_{i}} + q 1_{H}$ of [F3] would give $s_{i}^{2} = (q - 1) s_{i} + q 1$ in $C [S_{n}]$ ; since $s_{i}^{2} = 1$ (a transposition is an involution) this reads $(q - 1) s_{i} = (1 - q) 1$ , so $s_{i} = - 1$ because $q \neq 1$ . But $1$ and $s_{i}$ are linearly independent basis elements of $C [S_{n}]$ by [F4], so $s_{i} \neq - 1$ : a contradiction. Hence the Tits isomorphism cannot preserve the natural bases.

2.1F1step 1.1step 1.2

The isomorphism of step 1.1 is produced by the formal-lifting and constructible-incidence argument of Tits deformation, which selects no canonical basis and no prescribed bijection of simple modules; composing it with an algebra automorphism may change the induced bijection on simple modules, when equal-sized matrix factors are permuted; inner automorphisms leave simple isomorphism classes fixed, so no prescribed identification of the simple $H$ -modules with those of $C [S_{n}]$ is determined by the construction. What is invariant is exactly what step 1.1 records: the number and the dimensions of the simple modules. In particular the corollary asserts no natural bijection of simple modules.

3.1step 1.1step 1.2step 2.1∎

Step 1.1 gives the isomorphism and the numerical consequences, step 1.2 shows that no basis-preserving isomorphism exists, and step 2.1 records the non-canonicity. AC is inherited from the Tits-deformation supplier as declared, and all remaining objects are finite-dimensional over $C$ .

Depends on

Used by

The q=2 torus boundary Example
The trivial and Steinberg splitting on P¹(F_q) Example
The Tits isomorphism is noncanonical Remark
The constituents of the spherical principal series of GLₙ Theorem

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jay Taylor, Finite Reductive Groups - Corollary 5.19 and Remark 5.23, printed pp. 45-46 (standard reference, not scraped)
Ivan Losev, Lecture 8: Representations of GL_n(F_q) - Corollary 2.7 and the closing Remark on the natural bijection with $\operatorname{Irr}(S_n)$, PDF p. 5 (standard reference, not scraped)
Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Remark 11.6 and Theorem 11.14 (compatibility, not canonicity), printed pp. 47 and 51 (standard reference, not scraped)