Degenerate Ariki decomposition numbers in small rank

The four monomials and the label sets. The finite alphabet is $I^{+}=\{0,1\}$ and each charge-zero factor is its one-box natural module. Hence its tensor-square basis is $M_{00},M_{01},M_{10},M_{11}$. Weakly increasing rows select $00,01,11$, and weakly decreasing rows select $00,10,11$, so these are respectively $\mathrm{S}\mathrm{t}\mathrm{d}$ and $\mathrm{R}\mathrm{e}\mathrm{v}$; the mixed Bruhat order is $01<10$. The actual coordinate weights are $2{\delta }_0,{\delta }_0+{\delta }_1,{\delta }_0+{\delta }_1,2{\delta }_1$; subtracting each from the vacuum weight $2{\delta }_0$ gives contents $0,{\alpha }_0,{\alpha }_0,2{\alpha }_0$. In particular $I_{{\alpha }_0}=\{(0)\}$ and $I_{2{\alpha }_0}=\{(0,0)\}$: residue sequences have length equal to tower degree, independently of the two tensor factors.

The transition vectors and the reciprocal convention. On this mixed space the ordinary Fock bar is $\psi M_{01}=M_{01}$ and $\psi M_{10}=M_{10}+(s^{-1}-s)M_{01}$, with coefficient bar $s\mapsto s^{-1}$. Direct substitution shows that $L_{10}=M_{10}-s{M}_{01}$ and $T_{10}=M_{10}+s^{-1}{M}_{01}$ are bar fixed in their respective positive- and negative-power triangular charts; the endpoints and $L_{01}$ equal their monomials. Thus $M_{10}=L_{10}+s{L}_{01}$, and the ambient inverse-chart coefficient is $d_{01,10}=s$. Its row gives $P_{01}=M_{01}+s{M}_{10}=s{T}_{10}$. Under $t=s^{-1}$ these same formulas become $L_{10}=M_{10}-t^{-1}{M}_{01}$, $T_{10}=M_{10}+t{M}_{01}$ and $P_{01}=M_{01}+t^{-1}{M}_{10}$; reciprocating the parameter does not transpose the inverse transition matrix.

The normalized projection. The source coproduct gives $F_0{M}_{00}=M_{10}+s^{-1}{M}_{01}=T_{10}$ and $F_0{T}_{10}=(s+s^{-1})M_{11}$. Hence $T_{10}$ lies in the ground-generated constituent and is fixed by its normalized projection. The ground highest vector is $M_{00}$; $E_0{M}_{01}=M_{00}$, so $M_{01}$ is not highest. The same coproduct gives $E_0{M}_{10}=s{M}_{00}$, $F_0{M}_{01}=M_{11}$ and $F_0{M}_{10}=s{M}_{11}$, so both $E_0$ and $F_0$ kill $L_{10}$. Its weight is zero for $h_0$, hence it spans a trivial constituent, complementary to the three-dimensional ground constituent since $T_{10}$ and $L_{10}$ are linearly independent. Projection onto the ground constituent along this line is a module map fixing the ground, so this directly constructs the normalized projection and its off-standard kernel $\mathbb{Q}(s)L_{10}$. Since $T_{10}-L_{10}=(s+s^{-1})M_{01}$, applying the projection gives $D_{01}=\pi M_{01}=T_{10}\mathrm{/}(s+s^{-1})$ and $\pi M_{10}=s{D}_{01}$. The endpoints are fixed, and $\pi L_{10}=0$. At $s=1$ this is the half-sum $(M_{01}+M_{10})\mathrm{/}2$. All divisions here take place over $\mathbb{Q}(s)$; they do not identify the projected costandard lattice with the ambient integral monomial lattice.

The supported blocks. Set $x=x_1$, $y=x_2$, $u=s_1$. The defining relations of The degenerate cyclotomic Hecke algebra give $u^2=1$, $[x,y]=0$, $ux=yu-1$, $uy=xu+1$ and $x^2=0$ in the relevant degree; from $u{x}^2=0$ and the cross relations one obtains $y^{2}u=x+y$, hence $y^2=(x+y)u$, and reducing the relation $y^2=(x+y)u$ decreases the $y$-exponent while killing $x$-exponents at least $2$, so the eight words $1,x,y,xy,u,xu,yu,xyu$ span the degree-$2$ algebra. Map these generators to $(N,-N,-J)$ on $M_2(\mathbb{C})$ with $N=E_{12}$ and $J$ the swap matrix, to $(ϵ,1+ϵ,1)$ on $B_{+}=\mathbb{C}[ϵ]\mathrm{/}{ϵ}^2$, and to $(ϵ,-1+ϵ,-1)$ on $B_{-}=\mathbb{C}[ϵ]\mathrm{/}{ϵ}^2$; direct matrix computation verifies $u^2=1$, $[x,y]=0$, $x^2=0$, $ux=yu-1$ and $uy=xu+1$ in all three factors. The central element $z=x+y$ has images $0$, $1+2ϵ$ and $-1+2ϵ$, and the Hermite polynomials $E_0=(1-z^2{)}^2$, $E_{\pm }=(1-E_0\pm (3z-z^3)\mathrm{/}2)\mathrm{/}2$ map to the three separate identities; each factor is generated by its images of $x$ and $u$, so the map from the span of the eight words onto the eight-dimensional product is surjective and hence, by the spanning bound, an isomorphism. The $I=\{0\}$ spectral content cut keeps the factor $M_2(\mathbb{C})$ at content $2{\alpha }_0$ and discards the adjacent-content factors, so the supported degree-$2$ block is $M_2(\mathbb{C})$ with simple dimension $2$; degree one is $\mathbb{C}[x]\mathrm{/}{x}^2$ with simple dimension $1$ at content ${\alpha }_0$, and degree zero is $\mathbb{C}$. To verify vanishing independently of categorification, suppose an all-zero-residue supported module exists in degree at least three. Its restriction to the first two strands lies in the displayed $M_2$ factor, hence $x_2=-x_1$ and $x_1^2=0$. Since $s_2$ commutes with $x_1$, it also commutes with $x_2$. The affine recursion gives $x_3=s_2{x}_2{s}_2+s_2=x_2+s_2$. But a commuting sum of a square-zero operator and an involution has generalized eigenvalues only $1,-1$: on the two $s_2$ eigenspaces it is $\pm 1+x_2$, each invertible with inverse $\pm 1-x_2$. This contradicts the nilpotence required by residue zero for $x_3$. Every nonzero finite-dimensional algebra has its nonzero regular module, so no such supported algebra exists in degree three or higher.

The decomposition matrix by intrinsic modules. The intrinsic affine-induced Specht presentation in Schur functor images of standard, simple, projective and tilting objects gives, for either single-box label $01$ or $10$, the degree-one module $H_1\mathrm{/}{H}_{1}x=\mathbb{C}$; the two different singleton components have the same charge zero. For $11$, with one box in each component, it gives $H_2\mathrm{/}(H_{2}x+H_{2}y)$, since the inducing algebra is the polynomial algebra on the two strand variables and both act as zero on its one-dimensional inducing module. In the three factors of step 2.2, $x,y$ generate the whole left ideal in $B_{+}$ and $B_{-}$ because $y-x=\pm 1$; in the matrix factor $y=-x=-N$, so the quotient is $M_2\mathrm{/}{M}_{2}N$. Here $M_{2}N$ consists of matrices with first column zero, and the first-column map identifies the quotient with the defining simple ${\mathbb{C}}^2$. The degree-zero Specht is $\mathbb{C}$. Thus each of these four Spechts is the indicated simple, directly giving the displayed $3\times 4$ multiplicity matrix. Independently step 1.2 gives $d_{01,10}(1)=1$ and identity on standard columns, so these actual multiplicities agree with [F1]. There is one simple in each of the three supported blocks by step 2.2, so there is no additional simple $D_{10}$. The standard square submatrix is the identity; the rectangular matrix is not asserted invertible.

Conclusion. Steps 1.1-3.1 verify the transition, projection and decomposition statements, including the inverse-chart entry $d_{01,10}(1)=1$ in place of any negative transposed-inverse value, the identity on standard columns and the absence of an off-standard simple $D_{10}$; step 2.2 verifies the block dimensions $1,1,2$ at contents $0,{\alpha }_0,2{\alpha }_0$ of the supported tower and the vanishing of higher supported degrees. The two further adjacent-residue degree-$2$ blocks of the full algebra are outside this supported window and are treated in the companion showcase; no unprojected monomial is used as a simple class.