Type-A Hecke permutation modules: semistandard Murphy bases and Specht filtrations

The row-double-coset identity and membership. For a semistandard tableau $S$ of shape $\lambda$ and type $\rho$ and a standard $t$ of shape $\lambda$, every row-standard refinement $s$ of the column-strict tableau $S$ is standard, and ${\sum }_s{T}_{d(s)}^{\ast }{x}_{\lambda }$ is the sum of $T_w$ over the double coset $S_{\rho }d(s_0{)}^{-1}{S}_{\lambda }$, each $w$ occurring exactly once; its left $S_{\rho }$-coset expansion writes it as $x_{\rho }{h}_0$, and appending $T_{d(t)}$ gives $m_{(S,t)}=x_{\rho }h$ for some $h\in H_n$. Hence each $m_{(S,t)}$ lies in $x_{\rho }{H}_n$, and no row-intersection quantum factorial is introduced.

Independence and the Garnir coefficient rules. The sums $m_{(S,t)}$ are $R$-independent because their Murphy fibres (the index sets $\{s:\rho (s)=S\}$) are disjoint and every coefficient in the expansion of $m_{(S,t)}$ in the $m_{st}$ basis is $1$; they therefore have disjoint unit pivots. For the coefficient rules, consider a violation at adjacent rows $i,i+1$ and column $j$ and the composition $\nu =({\lambda }_1,\dots ,{\lambda }_{i-1},j-1,{\lambda }_i+1,{\lambda }_{i+1}-j,{\lambda }_{i+2},\dots )$; the two length-additive coset sums of $S_{\nu }{S}_{\lambda }$ give ${\sum }_{w\in S_{\nu }\cap D_{\lambda }}{x}_{\lambda }{T}_w={\sum }_{v\in S_{\lambda }\cap D_{\nu }}{T}_v^{\ast }{x}_{\nu },$ whose distinguished Garnir term occurs with coefficient $1$. Every other term in the original shape has strictly more dominant row tableaux, the reordered $\nu$ has strictly higher lexicographic shape, and the ordinary Murphy dominance refinement places these shape errors in higher-cell ideals; a nonstandard tableau factors as that Garnir tableau followed by an additive-length word (removing inversions while keeping the column violation), and Bruhat subword support preserves the strict tableau-prefix inequality. In the isolated relation for consecutive $a,a+1$ in one column, sorting the Garnir belt has a unique standard term whose single adjacent swap produces the distinguished Garnir term, with coefficient $-1$, and every other same-shape term is strictly more dominant. This yields the displayed congruence of the Statement, and the right analogue follows by the cellular involutions $\ast$.

The row eigenspace test. For a fixed $a$ the left $z$-eigenspace of $T_a$ in $H_n$ is $(1+T_a)H_n$: in the $T_w$ basis pair $w$ with $s_{a}w$ of larger length, and coefficient comparison in $T_{a}h=zh$ gives equality of the two coefficients, whose sum is $(1+T_a)T_w$. This argument uses only that $z$ is a unit. Over the universal domain, no element of this space has a nonzero Murphy coefficient at a tableau with $a,a+1$ in one column: choose a least shape and then a least left tableau in a dominance-compatible support order; higher-cell terms cannot contribute to its shape, and a column correction from another tableau can contribute only from a strictly less dominant left tableau, absent by minimality. If $a,a+1$ are in a column, its coefficient in $T_{a}h$ is the negative of its coefficient in $h$ by step 1.2, contradicting the eigenvalue equation since $z+1\mathrm{\ne }0$ in the domain; if they share a row, the basis element is already a $z$-eigenvector and is subtracted; if their swap is standard, the exact two-tableau rules make the two coefficients equal and their sum is a $z$-eigenvector, which is subtracted. Each subtraction stays in the row ideal and removes the chosen support term without creating new ones, so induction on the finite support proves the column exclusion; specializing the resulting coordinate-zero identities gives it over every base ring, with no division by $z+1$ at a singular parameter.

The basis theorem. Let $h\in x_{\rho }{H}_n$. It is a $z$-eigenvector for every generator inside every $\rho$-row interval, so step 2.1 excludes any supported tableau with a consecutive pair inside one column; for an adjacent standard swap, comparison of the exact two-tableau equations makes the two coefficients equal (no column-state correction can interfere, since every such state has coefficient $0$). The standard refinements of a fixed row-label tableau are the linear extensions of the finite node posets of its repeated-label classes, and any two are connected by swaps of adjacent incomparable elements; finite induction on the inversion distance shows that if a repeated label occupies one column, a consecutive vertical cover realises an excluded state and forces the whole fibre coefficient to be $0$, while otherwise the row-label tableau is column-strict and every fibre coefficient equals one common value. Hence $h$ has zero coefficients off semistandard fibres and constant coefficients on each semistandard fibre, i.e. $h$ lies in the span of the $m_{(S,t)}$; with step 1.2 this proves that the $m_{(S,t)}$ form an $R$-basis of $x_{\rho }{H}_n$.

The two filtrations, keeping the ordinary dictionary explicit. The higher-shape Murphy-cell filtration obtained from [F1] has one quotient $C_{\mathrm{M}}(\lambda )$ for every semistandard tableau $S$ of shape $\lambda$ and type $\rho$: its second-index action is the fixed-first-index Murphy cell action. These quotients are free over $R$, so every short exact sequence of this filtration splits as a sequence of $R$-modules. Star duality is therefore exact on this filtration. On the distinguished coset basis $b_d=x_{\rho }{T}_d$ of $x_{\rho }{H}_n$, define $\beta (b_d,b_c)={\delta }_{dc}{z}^{\mathrm{\ell }(d)}$. The row-permutation generator cases prove contravariance: a same-row generator acts by $z$; a length-increasing adjacent coset pair acts by $b_d{T}_i=b_c$, $b_c{T}_i=z{b}_d+(z-1)b_c$, with $\mathrm{\ell }(c)=\mathrm{\ell }(d)+1$, and the diagonal weights give $\beta (b_d{T}_i,b_c)=z^{\mathrm{\ell }(c)}=z\beta (b_d,b_d)$. All other entries are zero or the corresponding diagonal equality. Thus $\beta$ gives an $H_n$-isomorphism $x_{\rho }{H}_n\cong (x_{\rho }{H}_n{)}^{\circ }$, since every diagonal $z^{\mathrm{\ell }(d)}$ is a unit. The independently proved same-shape bridge gives $C_{\mathrm{M}}(\lambda {)}^{\circ }\cong P(\lambda )$, the ordinary Dipper–James Specht module, by finite-free biduality. Dualize the split Murphy filtration and transport it through this permutation self-duality. The resulting reversed filtration is an ordinary Specht filtration with exactly the same multiplicities. The original Murphy filtration is not called the ordinary one. This preserves the full every-ring/unit-parameter existence Statement without generic semisimplicity or form nondegeneracy on a Specht module.

The generic multiplicity interpretation. Over the generic field the finite Hecke algebra is split semisimple: by [F2] every simple is a nonzero cellular head, a simple factor of an ordinary cell module has a word character from that module and from the head's own cell shape, and disjoint word sets force the shapes to agree; existence of a factor forces its own head to be nonzero, diagonal multiplicity $1$ forces the cell module to be simple, and the sum of squared dimensions equals the Hecke rank, so the algebra and each parabolic are split semisimple. Consequently the filtration multiplicity $\mathrm{\#}{\mathrm{SStd}}_{\rho }(\lambda )$ equals the multiplicity of the trivial character of the $\rho$-row parabolic in $S^{\lambda }$, by induction/restriction adjunction in the semisimple category.

Conclusion. Steps 1.1, 1.2, 2.1 and 3.1 prove the semistandard Murphy basis of $x_{\rho }{H}_n$, step 1.2 gives the explicit Garnir and same-column generator rules with their termination, step 4.1 gives the Specht filtration with multiplicity $\mathrm{\#}{\mathrm{SStd}}_{\rho }(\lambda )$, and step 5.1 gives the generic parabolic-multiplicity interpretation. Everything is proved universally over $\mathbb{Z}[z,z^{-1}]$ and specializes; the row-double-coset membership uses each coset word once and introduces no quantum-factorial division.