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A block idempotent corner outside the stable range

Example

Assume the inherited Axiom of Choice. Take the pyramid $λ = (1, 1)$ , so $n = 2$ , $l = 1$ , $q_{1} = 2$ and $N = 2$ , and let $d = 3$ . The stable range $q_{l} \geq d$ fails, since $2 < 3$ ; the equivalent source condition is that at least $3$ parts of $λ$ are equal to $l = 1$ , which is false. With the origin $c = (0)$ one has $a_{1} = c_{1} + q_{1} - n = 0$ , so the level-one cyclotomic quotient is $H_{3} (Λ) = C [S_{3}]$ , of dimension $6$ , while the image $H_{3} (λ, c)$ in ${End}_{C} (V^{\otimes 3})^{o p}$ , $V = C^{2}$ , has dimension $5$ . The example identifies the image as a block idempotent corner $f C [S_{3}] f$ , verifying the central-corner statement of Stable range and block idempotent corner for higher-level Schur-Weyl duality in the smallest non-stable case.

Facts & Assumptions

Given: The pyramid $λ = (1, 1)$ , the natural ${g l}_{2}$ -module $V = C^{2}$ , the tensor space $V^{\otimes 3}$ , and the group algebra $C [S_{3}]$ .

[F1]

For this pyramid $l = 1$ , $q_{1} = 2$ , $e = 0$ , $g^{e} = {g l}_{2}$ and the level-one cyclotomic quotient is $H_{3} (Λ) = C [S_{3}]$ with its place-permutation action on $V^{\otimes 3}$ ; the stable range $q_{l} \geq d$ becomes $2 \geq 3$ , which fails (Stable range and block idempotent corner for higher-level Schur-Weyl duality, The finite W and degenerate cyclotomic Hecke actions satisfy a double centralizer).

[F2]

The $S_{3}$ -module $V^{\otimes 3}$ has a four-dimensional invariant space, with basis the symmetrizations of $v_{1}^{3}, v_{1}^{2} v_{2}, v_{1} v_{2}^{2}, v_{2}^{3}$ . Its sign space is zero by [F4]. By [F3] the remaining four dimensions are two copies of the two-dimensional standard module. Thus its group-algebra image is $C \times M_{2} (C)$ , of dimension five; classical Schur-Weyl duality identifies that image with the ${g l}_{2}$ commutant (The Schur-Weyl mutual centralizer theorem on tensor powers).

[F3]

Wedderburn decomposition $C [S_{3}] ≅ C \times C \times M_{2} (C)$ with the trivial, sign and standard blocks, indexed by the partitions $(3)$ , $(1, 1, 1)$ and $(2, 1)$ (If $k$ is algebraically closed and $char k ∤ ∣ G ∣$ , then $k [G] ≅ \prod_{i = 1}^{r} M_{n_{i}} (k)$ , Specht modules classify the complex irreducibles of $S_{n}$ ).

[F4]

For $\dim V = 2$ one has $Λ^{3} V = 0$ , so the antisymmetrizer $a = \sum_{w \in S_{3}} sgn (w) w$ acts on $V^{\otimes 3}$ as the antisymmetrization, which is $3! = 6$ times the projection onto $Λ^{3} V = 0$ ; hence $a$ acts by $0$ .

Proof

technique · direct

Given: The image $ψ_{3} (C [S_{3}]) \subseteq End (V^{\otimes 3})^{o p}$ and the Wedderburn blocks of [F3].

1.1F2F3algebra

By [F2], the tensor space contains the trivial $S_{3}$ module with multiplicity four, the standard module with multiplicity two, and no sign module. The decomposition [F3] therefore makes the group-algebra image exactly $C \times M_{2} (C)$ , of dimension five. By classical Schur-Weyl duality [F2] this is ${End}_{{g l}_{2}} (V^{\otimes 3})$ , whereas $\dim C [S_{3}] = 6$ .

2.1F3F4step 1.1algebra

The killed sign block. By [F3] the block decomposition of $C [S_{3}]$ is $C \times C \times M_{2} (C)$ ; the one-dimensional sign block is spanned by the antisymmetrizer $a$ of [F4], which acts on $V^{\otimes 3}$ by $0$ . Hence the kernel of $ψ_{3}$ contains the sign block; since the image has dimension $5$ by step 1.1, the kernel has dimension $1$ and is exactly the sign block. So $ψ_{3}$ kills precisely the sign block and is injective on the direct sum of the trivial and standard blocks.

3.1F3step 2.1algebra

The block idempotent corner. Let $f$ be the central idempotent of the trivial and standard blocks, so that $C [S_{3}] f = C [S_{3}] f^{2}$ is the corresponding two-sided ideal; its corner $f C [S_{3}] f$ has dimension $1 + 4 = 5$ by [F3]. Multiplication by $f$ , $x \mapsto f x f$ , is an algebra surjection $C [S_{3}] ↠ f C [S_{3}] f$ whose kernel contains the sign block and has dimension $6 - 5 = 1$ , hence is exactly the sign block; therefore $C [S_{3}] / ⁣ \ker ψ_{3} ≅ f C [S_{3}] f$ and the image $H_{3} (λ, c) = ψ_{3} (C [S_{3}])$ is isomorphic to the block idempotent corner $f C [S_{3}] f$ , which is the central-corner form $f H_{3} (a) f$ of the Definition.

4.1F1step 1.1step 3.1algebra∎

The stable-range check. Here $q_{l} = q_{1} = 2$ while $d = 3$ , so the stable range fails and the image is not the full algebra $H_{3} (a) = C [S_{3}]$ (which has dimension $6$ ); the corner of step 3.1 realizes the boundary case of the Definition, and the missing dimension $1$ is precisely the killed sign block.

Remarks

The example is the level-one, two-dimensional instance of the general corner statement: the stable range fails by one step, and the failure is visible as the dimension drop $6 - 5 = 1$ caused by the vanishing $Λ^{3} C^{2} = 0$ .
The nonadmissible origin $c = (0, 1)$ example of The finite W and degenerate cyclotomic Hecke actions satisfy a double centralizer is a different boundary: here the origin is admissible and the corner is a sum of entire central blocks, as required.

Depends on

Used by

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Sources

Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008) 1-57 (complete 50-page arXiv article) (standard reference, not scraped)
Jonathan Brundan and Alexander Kleshchev, The degenerate analogue of Ariki's categorification theorem, Math. Z. 266 (2010) 877-919 (complete 44-page arXiv article) (standard reference, not scraped)