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Vust's double centralizer in a small stable range

Example

Take the pyramid $λ = (1, 1)$ , so $n = 2$ , $l = 1$ , $q_{1} = 2$ , $N = 2$ , and $e = 0$ (all boxes lie in one column, and the grading is concentrated in degree $0$ ): $g = g^{e} = {g l}_{2} (C)$ , $m = 0$ , $C_{1} [x_{1}, \dots, x_{d}] = C [x_{1}, \dots, x_{d}] / (x_{1}, \dots, x_{d}) = C$ and $C_{1} [x] ⋊ C [S_{d}] = C [S_{d}]$ . The stable range for this $λ$ is $q_{l} \geq d$ , i.e. $2 \geq d$ , so $d = 1, 2$ are exactly the positive stable cases; degree zero is also stable. The example verifies Vust's double centralizer (Vust's graded double centralizer for the type-A nilpotent centralizer) in these cases.

Facts & Assumptions

Given: $V = C^{2}$ with the natural ${g l}_{2}$ -action, $e = 0$ , and $d = 1, 2$ .

[F1]

For $λ = (1, 1)$ the pyramid has $n = 2$ rows of length $1$ and one column of height $2$ ; the grading has $g_{0} = {g l}_{2}$ , $m = 0$ , and $e = 0$ (Vust's graded double centralizer for the type-A nilpotent centralizer, The type-A nilpotent pyramid, good grading and Whittaker character).

[F2]

The truncated variables vanish: $x_{i} = e = 0$ on $V^{\otimes d}$ , so $B = C [S_{d}]$ acting by place permutations (The degenerate cyclotomic Hecke algebra).

[F3]

The flip eigenspaces are $S^{2} V = span {v_{1} \otimes v_{1}, v_{1} \otimes v_{2} + v_{2} \otimes v_{1}, v_{2} \otimes v_{2}}$ and $Λ^{2} V = C (v_{1} \otimes v_{2} - v_{2} \otimes v_{1})$ . The latter is the determinant representation: $X$ acts by $tr X$ , not by zero. The three distinct $E_{11} - E_{22}$ weights on $S^{2} V$ , and the nonzero adjacent raising/lowering maps $E_{12}, E_{21}$ , show it is irreducible. The two summands are nonisomorphic. Classical Schur-Weyl duality identifies the full diagonal enveloping image with the commutant of the flip (The Schur-Weyl mutual centralizer theorem on tensor powers).

[F4]

$U ({g l}_{2}) \to End (V)$ is surjective, of dimension $4$ (Classical complex matrix Lie algebras, Representations of Lie algebras).

Proof

technique · direct

Given: The data and conventions of the Example.

1.1F4F2algebra

The case $d = 1$ . Here $B = C [S_{1}] = C$ , so ${End}_{{g l}_{2}} (V) = C \cdot 1$ by Schur's lemma and the image $φ_{1} (U ({g l}_{2})) = {End}_{C} (V)$ by [F4]; both Vust identities reduce to these two equal-dimensional statements.

1.2F3algebra

The case $d = 2$ , second centralizer. By [F3], $V^{\otimes 2} = S^{2} V \oplus Λ^{2} V$ with $Λ^{2} V$ one-dimensional and $S^{2} V$ three-dimensional irreducible. Since the flip acts by $+ 1$ on $S^{2} V$ and by $- 1$ on $Λ^{2} V$ , an endomorphism commuting with the $C [S_{2}]$ -action preserves each isotypic component, and its restrictions are arbitrary endomorphisms of the two (irreducible) components: ${End}_{C [S_{2}]} (V^{\otimes 2}) = End (S^{2} V) \oplus End (Λ^{2} V)$ , of dimension $9 + 1 = 10$ .

2.1F3step 1.2algebra

The case $d = 2$ , first centralizer. The image of $U ({g l}_{2})$ in $End (V^{\otimes 2})$ is the diagonal action on the two isotypic components, which by the mutual-centralizer statement of [F3] is the block-diagonal algebra $End (S^{2} V) \oplus End (Λ^{2} V)$ , also of dimension $10$ ; hence $φ_{2} (U ({g l}_{2})) = {End}_{C [S_{2}]} (V^{\otimes 2})$ . The other direction, ${End}_{{g l}_{2}} (V^{\otimes 2}) = C [S_{2}]$ , follows because $ψ_{2}$ is injective ( $\dim C [S_{2}] = 2$ and the flip is not a scalar on $V^{\otimes 2}$ ) and $\dim {End}_{{g l}_{2}} (V^{\otimes 2}) = 2$ by Schur's lemma applied to the two nonisomorphic components.

3.1F1F2step 1.1step 2.1∎

Range check. For $d \leq 2 = q_{l}$ the stable-range hypothesis $q_{l} \geq d$ holds, and for $d = 3$ it fails; this example therefore realizes the theorem exactly in its stable range and shows that the level-one case is the classical Schur-Weyl duality.

Remarks

In this example the truncated polynomial algebra degenerates to $C$ and the finite W-algebra degenerates to $U ({g l}_{2})$ ; this is the smallest situation in which the general theorem is visible without any deformation.
The companion counterexample with a nontrivial tensor module and a noninjective Hecke action is different: here the module is the raw tensor space with $e = 0$ , where the identities happen to hold in the stable range.

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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)