Ariki Categorification and Graded Decomposition Numbers — Examples

The algebra and its block. With $\xi =-1$ the relation $T_1^2=(\xi -1)T_1+\xi$ becomes $T_1^2=-2T_1-1$, i.e. $(T_1+1{)}^2=0$ with $t=T_1+1$ [F1]. The algebra $\mathbb{C}[t]\mathrm{/}(t^2)$ is local with radical $(t)$ and unique simple $D=\mathbb{C}$; its regular module is projective and indecomposable, so $P=H_2^{\mathrm{\Lambda }}$ 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 ${\alpha }_0+{\alpha }_1={\alpha }_1+{\alpha }_0$, which is therefore the only nonzero block and equals $\mathbb{C}[t]\mathrm{/}(t^2)$ [F1, F2].

The graded composition series. Multiplication by $t$ raises the degree by $\mathrm{deg}t=2$ in the transported grading, so $tP\cong D\{2\}$ and $P\cong D\oplus D\{2\}$ as graded vector spaces; hence the graded composition series is $0\subset tP\subset P$ with $[P]=(1+q^2)[D]$ in $G_0^{\mathrm{g}\mathrm{r}}(H_{{\alpha }_0+{\alpha }_1}^{\mathrm{\Lambda }})$, which is not a shift of $[D]$ [F2, F4].

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 $G^{\ast }(1,1)=(M_{1,1}+q{M}_{(2)})\mathrm{/}(1+q^2)$, so $G(1,1)=(1+q^2)G^{\ast }(1,1)$ [F3].

The identifications and the pairing. By [F4] the projective class $[P]$ corresponds to $G(1,1)$ and the simple class $[D]$ to $G^{\ast }(1,1)$, giving $[P]=(1+q^2)[D]\mapsto G(1,1)$ and $[D]\mapsto G^{\ast }(1,1)$ under the transported identifications. The graded Cartan pairing is read from ${\mathrm{Hom}}^0(P,D)=\mathbb{C}$ and $\mathrm{gdim}\mathrm{HOM}(P,P)=1+q^2$ (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].

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].

The polynomials and the class $[P]$. In $\mathbb{Z}[q,q^{-1}]$ the elements $1+q^2$ and $2$ differ (their difference $q^2-1$ is nonzero) while $(1+q^2){\mathrm{\mid }}_{q=1}=2$; multiplying by the nonzero class $[D]$ gives distinct classes $[P]=(1+q^2)[D]$ and $2[D]$ in the free $A$-module $G_0^{\mathrm{g}\mathrm{r}}(H_{{\alpha }_0+{\alpha }_1}^{\mathrm{\Lambda }})$, whose images under the forgetful map $\zeta$ (which satisfies $\zeta (qm)=\zeta (m)$) both equal $2[D]$ Graded module and projective Grothendieck groups of the KLR and Hecke towers [F1].

The Specht classes and the transition polynomials. The example computes $[S(1,1)]=[D]$ and $[S(2)]=q[D]$, which are distinct because $q\mathrm{\ne }1$; the forgetful map sends both to $[D]$. The corresponding transition polynomials are $d_{(1,1),(1,1)}(q)=1$ and $d_{(1,1),(2)}(q)=q$ in the recorded $e=2$ Fock computation, distinct in $\mathbb{Z}[q,q^{-1}]$ and equal at $q=1$ (Projective and simple Grothendieck classes in a small non-semisimple block, Level-one global-basis transition polynomials in the symmetric-group convention) [F2].

Conclusion. By the graded decomposition theorem the classes are read through the transition polynomials, so the failure above is exactly the statement that the specialization $q\mapsto 1$ loses the graded multiplicities and shift data (Graded decomposition numbers equal canonical-basis coefficients in characteristic zero). The displayed finite block and polynomial computations introduce no new choice; AC is inherited only through the invoked Fock/global-basis and graded-decomposition identifications.