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Cyclotomic Hecke algebras of level one are finite Hecke algebras

Statement

Let $Λ = Λ_{j}$ be a level-one fundamental weight and keep the conventions of Affine type-A presentations and charged cyclotomic quotients. Then the assignments $T_{r} \mapsto T_{r}$ ( $1 \leq r < d$ ) in the nondegenerate case and $s_{r} \mapsto s_{r}$ in the degenerate case extend to $F$ -algebra isomorphisms $H_{F, q} (S_{d}) ⟶ H_{d}^{Λ}, F S_{d} ⟶ H_{d}^{Λ}$ respectively (The generic type-A Hecke algebra, The standard basis of the generic type-A Hecke algebra).

Facts & Assumptions

Given: A field $F$ , the level-one fundamental weight $Λ = Λ_{j}$ , the cyclotomic quotient $H_{d}^{Λ}$ of Affine type-A presentations and charged cyclotomic quotients, and the finite type-A Hecke algebra $H_{F, q} (S_{d})$ with quadratic relation $T_{r}^{2} = (q - 1) T_{r} + q$ , $q = q_{H}$ , respectively the group algebra $F S_{d}$ .

[F1]

At level one the cyclotomic relation reads $X_{1} = q^{j}$ in the nondegenerate branch and $x_{1} = j \cdot 1_{F}$ in the degenerate branch; the mixed relations give $X_{r + 1} = q^{- 1} T_{r} X_{r} T_{r}$ and $x_{r + 1} = s_{r} x_{r} s_{r} + s_{r}$ , so every polynomial generator is expressible in the first one and the finite generators (Affine type-A presentations and charged cyclotomic quotients).

[F2]

The finite Hecke algebra has standard basis ${T_{w} : w \in S_{d}}$ ; the Jucys–Murphy elements $L_{r} = \sum_{a < r} (a r)$ in $F S_{d}$ commute and satisfy $s_{r} L_{r + 1} = L_{r} s_{r} + 1$ and $s_{r} L_{t} = L_{t} s_{r}$ for $t \notin {r, r + 1}$ , with the analogous multiplicative identities in $H_{F, q} (S_{d})$ (The Hecke Jucys-Murphy action is dominance triangular, The standard basis of the generic type-A Hecke algebra).

Proof

technique · direct

Given: The level-one quotient $H_{d}^{Λ}$ and the finite generators

$T_{r}$ (respectively $s_{r}$ ).

1.1F1algebra

Surjectivity of the natural map. By [F1] the first polynomial generator is the scalar $q^{j}$ (respectively $j$ ), and the recursion expressing every $X_{r + 1}$ (respectively $x_{r + 1}$ ) in $X_{r}$ and $T_{r}$ (respectively $s_{r}$ ) shows that the algebra generated by the finite generators and $X_{1}$ is already generated by the finite generators alone. Hence the assignment of the Statement is a surjective algebra homomorphism.

1.2F1F2construct

The inverse in the nondegenerate branch. Let $L_{r}$ be the additive Jucys–Murphy element and put $M_{r} : = 1 + (q - 1) L_{r} = q^{1 - r} T_{r - 1} \dots T_{1}^{2} \dots T_{r - 1},$ a product of units, so every $M_{r}$ is invertible; the $M_{r}$ commute because the $L_{r}$ do, and $T_{r} M_{r} T_{r} = q M_{r + 1}$ while $T_{r} M_{t} = M_{t} T_{r}$ for $t \notin {r, r + 1}$ follow from [F2] by separating the term $a = r$ in $L_{r + 1}$ and using the finite braid relation on the three affected transpositions. Setting $X_{r} : = q^{j} M_{r}$ therefore produces commuting units with $T_{r} X_{r} T_{r} = q X_{r + 1}$ , distant mixed commutation, the finite Hecke relations and $X_{1} = q^{j}$ . These elements satisfy the complete affine presentation, so the assignment defines a homomorphism $H_{d}^{Λ} \to H_{F, q} (S_{d})$ fixing the $T_{r}$ .

1.3F1F2construct

The inverse in the degenerate branch. Put $x_{r} : = j + L_{r}$ with $L_{r} = \sum_{a < r} (a r) \in F S_{d}$ . Direct transposition multiplication gives $s_{r} L_{r + 1} s_{r} = L_{r} + s_{r}$ , hence $s_{r} x_{r + 1} = x_{r} s_{r} + 1$ , and the $x_{r}$ commute and commute with distant generators by [F2]; at $r = 1$ this gives $x_{1} = j$ . These elements satisfy the degenerate affine relations, so the assignment defines a homomorphism $H_{d}^{Λ} \to F S_{d}$ fixing the $s_{r}$ .

2.1step 1.1step 1.2step 1.3algebra∎

Mutual inverses. The two constructions of steps 1.2 and 1.3 fix every finite generator, and the composites with the surjection of step 1.1 fix every presentation generator: the finite generators by construction, and the polynomial generators because both sides satisfy the same recursion from $X_{1} = q^{j}$ (respectively $x_{1} = j$ ) and the mixed relations. Hence both composites are the identity on a generating set, so the maps are mutually inverse isomorphisms; rank zero gives $F$ on both sides. This removes any late-basis or dimension dependency and introduces no BK or primitive-block input.

Remarks

The identification is proved by an explicit inverse rather than by comparing dimensions, so it does not depend on the later cyclotomic basis theorem, the BK isomorphism or any block-primitivity statement. No choice principle is used.

Depends on

Used by

Dependency tree · two levels

17 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

Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, arXiv:0808.2032v3, Invent. Math. 178 (2009) 451–484 (complete 32-page article) (standard reference, not scraped)
Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, arXiv:0909.4844v1, Bull. AMS 47 (2010) 419–481 (complete 66-page survey) (standard reference, not scraped)
Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete prepublication manuscript (AMS University Lecture Series 15, 1999), 79 pages (standard reference, not scraped)