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Projective and simple Grothendieck classes in a small non-semisimple block

Example

Take $F = C$ , $ξ = - 1$ (so $e = 2$ ), level one $Λ_{0}$ and $d = 2$ . Then $H_{2}^{Λ} = H_{C, - 1} (S_{2})$ with $T_{1}^{2} = - 2 T_{1} - 1$ , i.e. $(T_{1} + 1)^{2} = 0$ ; the algebra is $C [t] / (t^{2})$ with $t = T_{1} + 1$ , a local algebra whose unique simple is $D = C$ with $t$ acting as $0$ , and whose regular module is the unique indecomposable projective $P = H_{2}^{Λ}$ . Both partitions of $2$ have content $α_{0} + α_{1}$ in the charge- $0$ convention, so $α_{0} + α_{1}$ is the only nonzero block and $H_{α_{0} + α_{1}}^{Λ} = C [t] / (t^{2})$ ; its transported grading has $\deg t = 2$ (the KLR block is the rank-two $e = 2$ block $F [t] / (t^{2})$ with $\deg y_{2} = 2$ ). The graded Specht modules are the one-dimensional modules $S (2) = C$ and $S (1, 1) = C$ , both with trivial underlying action of $t$ , and the graded composition series of $P$ is $0 \subset t P \subset P$ with $t P ≅ D {2}$ , so $[P] = (1 + q^{2}) [D] in G_{0}^{g r} (H_{α_{0} + α_{1}}^{Λ}), [P] \notin {[D], q^{\pm 1} [D], \dots} .$ The restricted label is $(1, 1)$ (since $(1, 1)$ is $2$ -restricted and $(2)$ is not), the edge degrees of the two shapes are $\deg (S (1, 1)) = 0$ and $\deg (S (2)) = 1$ , so the graded Specht classes are $[S (1, 1)] = [D]$ and $[S (2)] = q [D]$ , and the level-one Fock data give $G (1, 1) = M_{1, 1} + q M_{(2)}$ and $G^{*} (1, 1) = (M_{1, 1} + q M_{(2)}) / (1 + q^{2})$ ; hence under the transported identifications of Ariki's canonical basis theorem in characteristic zero $[P] ⟼ G (1, 1) = M_{1, 1} + q M_{(2)} = (1 + q^{2}) G^{*} (1, 1), [D] ⟼ G^{*} (1, 1) .$ The graded Cartan pairing of The graded Cartan pairing between projective and module classes has $h ([P], [D]) = 1$ and $h ([P], [P]) = 1 + q^{2}$ , while the graded composition multiplicity is $[P : D]_{q} = 1 + q^{2}$ (ungraded value $2$ ). Thus the projective class $[P]$ is not a simple class, and the two bases are dual to each other under $h$ without being equal as classes. All computations are finite and explicit; choice is inherited only through the transported identifications cited.

Facts & Assumptions

Given: The field $C$ , the parameter $ξ = - 1$ , the level-one fundamental weight $Λ_{0}$ , and $d = 2$ .

[F1]

The level-one Hecke algebra is the finite algebra $H_{F, ξ} (S_{2})$ with $T_{1}^{2} = (ξ - 1) T_{1} + ξ$ , and its residue-content blocks are the $e_{α} H_{d}^{Λ}$ (prop-cyclotomic-hecke-algebras-of-level-one-are-finite-hecke-algebras, def-residue-sequence-idempotents-and-cyclotomic-hecke-blocks).

[F2]

The transported grading (nontrivial for this $e = 2$ rank-two block, the corollary's explicit witness) and the blockwise KLR identification are supplied by cor-cyclotomic-hecke-blocks-inherit-a-nontrivial-z-grading; the KLR block satisfies the Hecke relations (lem-cyclotomic-klr-block-satisfies-the-hecke-relations).

[F3]

The graded Specht modules, their degree recursion and the degree of a standard tableau are those of The graded Specht module by homogeneous standard tableaux; the level-one transition data in the $e = 2$ case are the recorded computation of Level-one global-basis transition polynomials in the symmetric-group convention.

[F4]

The graded Grothendieck groups, their shift action and the graded Cartan pairing $h$ are those of Graded module and projective Grothendieck groups of the KLR and Hecke towers and The graded Cartan pairing between projective and module classes; the projective and simple classes are matched with $G (1, 1)$ and $G^{*} (1, 1)$ by Ariki's canonical basis theorem in characteristic zero, and the graded multiplicities by Graded decomposition numbers equal canonical-basis coefficients in characteristic zero.

Proof

technique · direct

1.1F1F2

The algebra and its block. With $ξ = - 1$ the relation $T_{1}^{2} = (ξ - 1) T_{1} + ξ$ becomes $T_{1}^{2} = - 2 T_{1} - 1$ , i.e. $(T_{1} + 1)^{2} = 0$ with $t = T_{1} + 1$ [F1]. The algebra $C [t] / (t^{2})$ is local with radical $(t)$ and unique simple $D = C$ ; its regular module is projective and indecomposable, so $P = H_{2}^{Λ}$ is the unique indecomposable projective [F1]. The nodes of $(2)$ have residues $0, 1$ and those of $(1, 1)$ have residues $0, - 1 \equiv 1$ , so both partitions lie in the block of content $α_{0} + α_{1} = α_{1} + α_{0}$ , which is therefore the only nonzero block and equals $C [t] / (t^{2})$ [F1, F2].

1.2F2F4

The graded composition series. Multiplication by $t$ raises the degree by $\deg t = 2$ in the transported grading, so $t P ≅ D {2}$ and $P ≅ D \oplus D {2}$ as graded vector spaces; hence the graded composition series is $0 \subset t P \subset P$ with $[P] = (1 + q^{2}) [D]$ in $G_{0}^{g r} (H_{α_{0} + α_{1}}^{Λ})$ , which is not a shift of $[D]$ [F2, F4].

1.3F3

Specht degrees and Fock data. The unique tableau of shape $(2)$ has degree $d_{A} (2) = 1$ (the addable $1$ -node $(2, 1)$ lies strictly below the removable node $(1, 2)$ , and there is no removable $1$ -node below it), and the unique tableau of $(1, 1)$ has degree $0$ ; hence $[S (2)] = q [D]$ and $[S (1, 1)] = [D]$ [F3]. The recorded $e = 2$ Fock computation gives $G (1, 1) = M_{1, 1} + q M_{(2)}$ and $G^{*} (1, 1) = (M_{1, 1} + q M_{(2)}) / (1 + q^{2})$ , so $G (1, 1) = (1 + q^{2}) G^{*} (1, 1)$ [F3].

2.1F3F4step 1.2

The identifications and the pairing. By [F4] the projective class $[P]$ corresponds to $G (1, 1)$ and the simple class $[D]$ to $G^{*} (1, 1)$ , giving $[P] = (1 + q^{2}) [D] \mapsto G (1, 1)$ and $[D] \mapsto G^{*} (1, 1)$ under the transported identifications. The graded Cartan pairing is read from ${Hom}^{0} (P, D) = C$ and $gdim HOM (P, P) = 1 + q^{2}$ (endomorphisms are the identity and multiplication by $t$ , of degree $2$ ), so $h ([P], [D]) = 1$ and $h ([P], [P]) = 1 + q^{2}$ ; the graded composition multiplicity is $[P : D]_{q} = 1 + q^{2}$ with ungraded value $2$ [F4]. Hence the projective class is not a simple class, and the projective and simple families are dual but not equal [F4].

3.1F4given∎

Choice. The computation uses the explicit two-dimensional algebra, its two partitions and the recorded $e = 2$ polynomials; no family of nonempty sets is selected, so choice is inherited only through the transported identifications cited in [F4].

Depends on

Used by

Specializing the grading variable to one forgets graded decomposition information Counterexample

Dependency tree · two levels

75 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, Graded decomposition numbers for cyclotomic Hecke algebras, arXiv:0901.4450v3, Adv. Math. 222 (2009) 1883-1942 (complete 57-page article) (standard reference, not scraped)
Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, arXiv:0909.4844v1, Bull. Amer. Math. Soc. 47 (2010) 419-481 (complete 66-page survey) (standard reference, not scraped)