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Charge reversal and the Fock involution in a small example

Example

Assume the Axiom of Choice inherited through the categorification and sign/Young suppliers; the explicit eight-dimensional calculations below use only finite-dimensional linear algebra.

Take $I = Z$ , level $l = 2$ and charge $(0, 0)$ , so $Λ = 2 Λ_{0}$ with $Λ^{t} = Λ$ . Use the semi-infinite columns

$O = (0, - 1, - 2, \dots), U = (1, - 1, - 2, \dots), R = (2, - 1, - 2, \dots), C = (1, 0, - 2, - 3, \dots),$

corresponding to the partitions $\emptyset, 1, 2, 11$ . The FULL height- $\leq 2$ ambient window consists of eight monomials

$00 = O O; 01 = O U; 10 = U O; 11 = U U; 0 R = O R; R 0 = R O; 0 C = O C; C 0 = C O .$

Their root degrees are $0$ ; $α_{0}$ for $01, 10$ ; $2 α_{0}$ for $11$ ; $α_{0} + α_{1}$ for $0 R, R 0$ ; and $α_{0} + α_{- 1}$ for $0 C, C 0$ . Here $S t d = {00, 01, 11, 0 R, 0 C}$ and $R e v = {00, 10, 11, R 0, C 0}$ ; rectification maps $10 \mapsto 01$ , $R 0 \mapsto 0 R$ , $C 0 \mapsto 0 C$ , and fixes $00, 11$ . Multipartition transpose reverses components and transposes their shapes:

$01 \leftrightarrow 10, 0 R \leftrightarrow C 0, R 0 \leftrightarrow 0 C; 00, 11 fixed .$

This is an ambient monomial permutation, not a permutation of nonexistent Hecke simple or projective labels off $S t d$ .

At parameter one the full algebra window is $H_{0} = C$ , $H_{1} = C [x] / x^{2}$ and $H_{2} = M_{2} (C) \oplus B_{+} \oplus B_{-}$ with $B_{\pm} = C [ϵ] / ϵ^{2}$ , the respective degree-two contents being $2 α_{0}$ , $α_{0} + α_{1}$ and $α_{0} + α_{- 1}$ . Sign sends $x_{r} \mapsto - x_{r}$ , $s_{r} \mapsto - s_{r}$ ; it fixes the unique $α_{0}$ simple and cover, fixes the unique $2 α_{0}$ simple and cover, and exchanges the simple/cover pairs in $B_{+}$ and $B_{-}$ . On $S t d$ labels it fixes $00, 01, 11$ and swaps $0 R \leftrightarrow 0 C$ . Equivalently the simple-label map is $B \mapsto (B^{↑})^{t}$ , not $B \mapsto B^{t}$ without rectification, and $D_{10} = D_{R 0} = D_{C 0} = 0$ .

The projected simple vectors are $D_{01} = \frac{1}{2} (M_{01} + M_{10})$ , $D_{0 R} = \frac{1}{2} (M_{0 R} + M_{R 0})$ , $D_{0 C} = \frac{1}{2} (M_{0 C} + M_{C 0})$ , $D_{00} = M_{00}$ , $D_{11} = M_{11}$ ; their projective cover vectors are $P_{01} = M_{01} + M_{10}$ , $P_{0 R} = M_{0 R} + M_{R 0}$ , $P_{0 C} = M_{0 C} + M_{C 0}$ , with the endpoints equal to $D$ . The ambient transpose at parameter one descends to the highest constituent to fix $P_{01} / D_{01}$ and to exchange $P_{0 R} / D_{0 R}$ with $P_{0 C} / D_{0 C}$ ; it does NOT swap $P_{01}$ with the ambient $P_{10}$ , nor $D_{01}$ with $D_{10}$ .

For the full Young namespace of Sign duality and Ringel duality match the Fock-space involutions the block models give

$Y_{00} = C, Y_{01} = B_{0}, Y_{10} = D_{01}, Y_{11} = D_{11}, Y_{0 R} = B_{+}, Y_{R 0} = D_{0 R}, Y_{0 C} = B_{-}, Y_{C 0} = D_{0 C},$

${\tilde{Y}}_{00} = C, {\tilde{Y}}_{01} = D_{01}, {\tilde{Y}}_{10} = B_{0}, {\tilde{Y}}_{11} = D_{11}, {\tilde{Y}}_{0 R} = D_{0 R}, {\tilde{Y}}_{R 0} = B_{+}, {\tilde{Y}}_{0 C} = D_{0 C}, {\tilde{Y}}_{C 0} = B_{-},$

where $B_{0} = C [x] / x^{2}$ . Thus cover labels are $S t d$ for the Young family and $R e v$ for the signed family; the off- $S t d$ Young images and off- $R e v$ signed images remain in the ambient family and need not be Hecke projectives. For $A \in R e v$ one has $\tilde{Y} (A) = σ^{*} Y (A^{t}) ≅ Y (A^{↓})$ , exactly the charge-reversal/rectification formula. Every displayed Specht is self-dual in these blocks, and $σ^{*} S (A^{t}) ≅ S (A)^{⊛}$ ; the two adjacent content blocks are essential to this showcase. In the general asymmetric case the source charge formula remains $Λ^{t} = Λ_{- m_{l}} + \dots + Λ_{- m_{1}}$ , and the example does not identify that tower with its original one.

Facts & Assumptions

Given: The global sign/transpose proposition Sign duality and Ringel duality match the Fock-space involutions and the categorification dictionary of The degenerate Ariki categorification theorem, with the Fock columns and monomials of The higher-level wedge-tensor Fock module and The dual-canonical, canonical and quasi-canonical Fock bases. We verify the label enumeration, the height-two algebra, the projected vectors, the sign permutation and the full Young/signed tables, under that inherited choice scope.

[A1]

The Axiom of Choice (The Axiom of Choice) is inherited solely from the local Whittaker, projective/tilting and categorification suppliers used for the projected-vector and full Young dictionary; the block presentations, matrices, finite basis enumeration and explicit sign twists require only finite choice.

[F1]

For $I = Z$ the sign map $σ (s_{r}) = - s_{r}$ , $σ (x_{r}) = - x_{r}$ gives $H_{α}^{Λ} ≅ H_{α^{t}}^{Λ^{t}}$ , intertwines residue $i$ with $- i$ , and satisfies $σ^{*} S (A^{t}) ≅ S (A)^{⊛}$ and $σ^{*} D (A^{t}) ≅ D (A^{↓})$ on reverse-standard labels; the multipartition transpose is the reversed-component shape transpose (Sign duality and Ringel duality match the Fock-space involutions).

[F2]

The simple and projective labels are indexed by the standard tableaux, with $D_{A} = π L_{A}$ for $A \in {S t d}^{Λ}$ and zero for the off-standard classes; the four Fock families $M_{A}, L_{A}, T_{A}, P_{A}$ with the quasi-canonical expansion $P_{A} = \sum_{B} d_{A, B} (s) M_{B}$ are the source dictionary used throughout (The degenerate Ariki categorification theorem).

Verification

technique · direct

1.1givenalgebra

The eight ambient labels. There is one bipartition of size zero, two of size one and five of size two: $(\emptyset, \emptyset)$ ; $(\emptyset, 1), (1, \emptyset)$ ; and $(1, 1), (\emptyset, 2), (2, \emptyset), (\emptyset, 11), (11, \emptyset)$ . The ground-difference column rule at charge $(0, 0)$ gives respectively $O O, O U, U O, U U, O R, R O, O C, C O$ , so $C 0$ has the vertical partition in the first component and $0 C$ in the second. Subtracting the ground weights gives degree $0$ for $00$ , $α_{0}$ for $01, 10$ , $2 α_{0}$ for $11$ , $α_{0} + α_{1}$ for $0 R, R 0$ , and $α_{0} + α_{- 1}$ for $0 C, C 0$ . Strict columns with weakly increasing rows select $00, 01, 11, 0 R, 0 C$ ; reverse-standard rows select $00, 10, 11, R 0, C 0$ . Reading the two columns and inserting in this small window gives $10^{↓} = 01$ , $R 0^{↓} = 0 R$ , $C 0^{↓} = 0 C$ and fixes $00, 11$ . Component reversal and partition transpose give precisely the displayed ambient permutation.

2.1step 1.1givenalgebra

The full height-two window. With $x = x_{1}$ , $y = x_{2}$ , $u = s_{1}$ the relations give $u^{2} = 1$ , $[x, y] = 0$ , $u x = y u - 1$ , $u y = x u + 1$ and $x^{2} = 0$ ; from $u x^{2} = 0$ one derives $y^{2} = (x + y) u$ , and reducing by this relation decreases the $y$ -exponent while killing $x$ -exponents at least two, so the eight words $1, x, y, x y, u, x u, y u, x y u$ span the degree-two algebra. The three models $(N, - N, - J)$ on $M_{2} (C)$ with $N = E_{12}$ and $J$ the swap matrix, $(ϵ, 1 + ϵ, 1)$ on $B_{+}$ , and $(ϵ, - 1 + ϵ, - 1)$ on $B_{-}$ satisfy the relations by direct matrix computation; the central element $z = x + y$ has images $0$ , $1 + 2 ϵ$ and $- 1 + 2 ϵ$ , and the Hermite polynomials $E_{0} = (1 - z^{2})^{2}$ , $E_{\pm} = (1 - E_{0} \pm (3 z - z^{3}) / 2) / 2$ map to the three separate identities. Each factor is generated by its images of $x$ and $u$ , so the map is surjective onto the eight-dimensional product and the spanning bound makes it an isomorphism; hence there are no other height-two blocks and $H_{0}, H_{1}$ are the direct factors listed. The two-dimensional simple of $M_{2} (C)$ carries the generator-fixed contravariant form $J$ , since $N^{t} J = J N$ and $(- J)^{t} J = J (- J)$ ; sign is inner on $M_{2} (C)$ via conjugation by $K = diag (1, - 1)$ (which satisfies $K N = - N K$ and $K J = - J K$ ), so the unique simple and its cover are fixed there, while on $B_{\pm}$ sign identifies $ϵ$ with $- ϵ$ and exchanges the two factors, fixing the unique objects of those contents as classes.

3.1A1step 1.1step 2.1F1F2givenalgebra

The projected vectors and the transpose. Normalizing the highest pairing $C_{0}$ makes the two-term highest sums have norm two, so the dual projected vectors are the half-sums $D_{01} = \frac{1}{2} (M_{01} + M_{10})$ , $D_{0 R} = \frac{1}{2} (M_{0 R} + M_{R 0})$ , $D_{0 C} = \frac{1}{2} (M_{0 C} + M_{C 0})$ , and the projective cover vectors are the unhalved sums with endpoints $P_{00} = D_{00}$ , $P_{11} = D_{11}$ . Applying the mixed residue operators $f_{1}$ and $f_{- 1}$ to the mixed highest vector gives $T_{R 0} = M_{R 0} + s^{- 1} M_{0 R}$ and $T_{C 0} = M_{C 0} + s^{- 1} M_{0 C}$ , with kernels $L_{R 0} = M_{R 0} - s M_{0 R}$ and $L_{C 0} = M_{C 0} - s M_{0 C}$ , so at parameter one these kernel vectors have zero projected image and the projected vectors are the displayed half-sums. The ambient monomial transpose at parameter one fixes the ground, commutes with the reflected Chevalley operators (transposing each column negates its contents and the coproduct is cocommutative), and therefore descends to the highest constituent, where it fixes $P_{01} / D_{01}$ and exchanges $P_{0 R} / D_{0 R}$ with $P_{0 C} / D_{0 C}$ ; it does not swap $P_{01}$ with the ambient $P_{10}$ , and the off-standard vectors $D_{10}, D_{R 0}, D_{C 0}$ are zero by [F2]. This is the projected-lattice clause of [F1] on the eight-label window.

3.2step 2.1F1givenalgebra

The intrinsic Young and signed images. Verify the tables from the intrinsic ideals rather than assuming a projective-preservation theorem. In $B_{0} = C [x] / x^{2}$ the reduced permutation ideal of the label $01$ is $B_{0}$ , and of the reverse label $10$ it is $(x)$ ; in $B_{\pm}$ the reduced permutation ideals are $B_{\pm}$ or $(x y) = (\pm ϵ)$ , where the row factor $1 \pm u$ equals $2$ when it occurs; in $M_{2} (C)$ both labels $11$ give the left ideal $M_{2} N$ , the unique simple cover. Every such ideal is indecomposable in its explicit block, so the intrinsic Young definition identifies them exactly: $Y_{00} = C$ , $Y_{01} = B_{0}$ , $Y_{10} = D_{01}$ , $Y_{11} = D_{11}$ , $Y_{0 R} = B_{+}$ , $Y_{R 0} = D_{0 R}$ , $Y_{0 C} = B_{-}$ , $Y_{C 0} = D_{0 C}$ , with cover labels exactly $S t d$ . The signed reduced permutation ideals of the same labels give ${\tilde{Y}}_{00} = C$ , ${\tilde{Y}}_{01} = D_{01}$ , ${\tilde{Y}}_{10} = B_{0}$ , ${\tilde{Y}}_{11} = D_{11}$ , ${\tilde{Y}}_{0 R} = D_{0 R}$ , ${\tilde{Y}}_{R 0} = B_{+}$ , ${\tilde{Y}}_{0 C} = D_{0 C}$ , ${\tilde{Y}}_{C 0} = B_{-}$ , with cover labels exactly $R e v$ ; no higher summand or nonexistent off-standard simple is needed.

4.1step 2.1step 3.2F1algebra

The sign formula on $R e v$ . For $A = 10, R 0, C 0$ , respectively, $A^{t} = 01, 0 C, 0 R$ and $A^{↓} = 01, 0 R, 0 C$ . By the tables of step 3.2, $Y (A^{t}) = B_{0}, B_{-}, B_{+}$ ; the sign action of step 2.1 sends these to $B_{0}, B_{+}, B_{-}$ , respectively. These are both $\tilde{Y} (A)$ and $Y (A^{↓})$ . For $A = 00, 11$ all three classes agree and are fixed by sign. Thus $\tilde{Y} (A) = σ^{*} Y (A^{t}) ≅ Y (A^{↓})$ holds for every reverse-standard label in the window. Each displayed Specht is self-dual under the generator-fixed duality in its block: it is one dimensional in $B_{0}, B_{\pm}$ , while the matrix-block simple has the explicit form $J$ of step 2.1. The transpose exchanges the two adjacent contents, and their one-dimensional generator actions are carried into each other by sign, proving the Specht formula here as well.

5.1step 3.1step 4.1given∎

Conclusion. Step 1.1 verifies the full eight-label ambient window and the rectification; step 2.1 proves the complete height-two algebra, hence the exact block models and the sign action; step 3.1 verifies the projected simple and cover vectors and the effect of the ambient transpose on the highest constituent, including its failure to swap $P_{01}$ with $P_{10}$ ; steps 3.2 and 4.1 verify the full Young and signed tables from the intrinsic ideals and the formula $\tilde{Y} (A) = σ^{*} Y (A^{t}) ≅ Y (A^{↓})$ on $R e v$ . The example covers the complete height- $\leq 2$ sign window; the general asymmetric case keeps the source formula $Λ^{t} = Λ_{- m_{l}} + \dots + Λ_{- m_{1}}$ and does not identify the reflected tower with the original one.

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Sources

Brundan-Kleshchev, The degenerate analogue of Ariki's categorification theorem (complete 44-page body, matching native fetch stamp) (standard reference, not scraped)