Cyclotomic semistandard row-type sums give permutation-module bases

The affine expansion with only variable-difference denominators. Writing $T_w$ as the product of the ${\mathrm{\Phi }}_i$ and the scalars $c_{i,i+1}$ along a reduced word of $w$ and moving a Laurent polynomial $f$ leftward across the $\mathrm{\Phi }$'s by [F2] expresses $T_{w}f$ as a sum of terms $c_{v}v(f)$ times a monomial in a subword, where $v$ runs over products of subwords of that reduced word, every denominator of $c_v$ is a product of differences $X_i-X_j$, and each $v$ is Bruhat-below $w$ by the subword characterization of Bruhat order ([F6]). In particular applying this with $f=u_{\lambda }$ keeps all denominators among the variable differences and introduces no $U_a-U_b$, quantum factorial, or Gram determinant.

Flagging is Bruhat-downward closed and gives monic divisibility. Let $S$ be semistandard of shape $\lambda$ and type $\mu$, and for a permutation $\sigma$ let $\pi ={\sigma }^{-1}$ map each label to its physical position; set $A_k={\sum }_{a<k}\mathrm{\mid }{\lambda }^{(a)}\mathrm{\mid }$ and $B_k={\sum }_{a<k}\mathrm{\mid }{\mu }^{(a)}\mathrm{\mid }$. By [F1] the flag condition says that a label of component $a$ occupies only a physical component $b\le a$, i.e. $\mathrm{\#}\{r\le B_k:\pi (r)\le A_k\}=B_k$ for all $k$. The count is at most $B_k$; inversion preserves the Bruhat order and a Bruhat-smaller permutation has at least as many entries in every prefix rectangle, so every Bruhat-smaller permutation of a flagged one is flagged. If $\sigma$ is flagged then $\sigma (u_{\lambda })={\prod }_k{\prod }_{j\le A_k}(X_{\sigma (j)}-U_k)$ contains the factor $(X_r-U_k)$ for every $r\le B_k$, because $\pi (r)\le A_k$; hence $\sigma (u_{\lambda })=u_{\mu }{\prod }_j{\prod }_{a(j)<k\le b(j)}(X_j-U_k)$ with $a(j)$ the physical component of the position occupied by label $j$ and $b(j)$ its type component, a monic polynomial. In particular $u_{\mu }$ divides $\sigma (u_{\lambda })$ in the Laurent polynomial ring.

The finite action and its flagged restriction. Work over the fraction field $K$ of the universal ring. Put $J_{\lambda }={\prod }_b{S}_{\mathrm{\mid }{\lambda }^{(b)}\mathrm{\mid }}$, the subgroup preserving the component intervals, and let $V_b$ be the ordinary left Murphy cell of shape ${\lambda }^{(b)}$. The shape-$\lambda$ left cell, restricted to the finite Hecke algebra, is $H_K(S_n){\otimes }_{H_K(J_{\lambda })}{⨂}_b{V}_b$. To verify this, factor each standard multitableau into its increasing component shuffle and its internal standard tableaux; the corresponding coset words are length-additive. The map sending the shuffle tensor these internal basis vectors to the corresponding left Murphy vector is a bijection of bases. It intertwines each finite generator: a generator inside a component gives the ordinary cell rule [F4]; a generator exchanging two components gives the two-term standard-word rule [F5]. An internal column failure uses exactly the ordinary Garnir straightening, with its component shuffle factored out. This is the same finite-generator calculation as the cellular proof, so it establishes the tensor model without generic rank inference. Now restrict to the $\mu$-row subgroup $P=S_{\mu }$. Its double cosets with $J_{\lambda }$ are indexed by matrices $N_{r,b}$ counting labels from $\mu$-row $r$ allocated to physical component $b$, with row sums the $\mu$-row lengths and column sums $\mathrm{\mid }{\lambda }^{(b)}\mathrm{\mid }$. In the standard-word basis, $H_K(P)T_d{H}_K(J_{\lambda })$ for a minimal double-coset word $d$ is stable under both parabolics. Length-additive multiplication identifies its induced cell subspace with induction from the intersection parabolic; this intersection splits the $r$-th $\mu$-row into consecutive pieces of sizes $N_{r,b}$, after the increasing relabelling. Indeed $T_i{T}_d=T_d{T}_j$ for each paired internal adjacent generator, and the coset words times internal basis vectors give the bijection. Hence the restricted left cell is the direct sum of these allocation subspaces. The flagged left cell consists of exactly the summands for which $N_{r,b}=0$ whenever $b$ exceeds the type component of row $r$: every tableau in that summand has precisely this allocation, and the converse is the flag condition. The full left cell includes the unflagged allocations as well.

The left row-double-coset identity and localized membership. Let $s_0$ be the least standard refinement of $S$ and $D=S_{\mu }d(s_0{)}^{-1}{S}_{\lambda }$. Every row-standard refinement of the semistandard $S$ is standard, and the ordinary length-additive left row-double-coset expansion gives ${\sum }_{s:\mu (s)=S}{T}_{d(s)}^{\ast }{x}_{\lambda }={\sum }_{w\in D}{T}_w$, each element of $D$ exactly once, with no row-intersection quantum factorial ([F4] proves the identical finite identity; [F1] fixes the elements $T_{d(s)}^{\ast }$ and $x_{\lambda }$). Since $S_{\mu }$ permutes labels inside $\mu$-components and $S_{\lambda }$ permutes positions inside $\lambda$-rows, every $w\in D$ is flagged; hence every Bruhat-subword permutation $v$ in the expansion of step 1.1 is flagged by step 1.2, so $v(u_{\lambda })=u_{\mu }{g}_v$ with $g_v$ a Laurent polynomial, and $T_w{u}_{\lambda }\in u_{\mu }{H}_n^{\mathrm{a}\mathrm{f}\mathrm{f}}[(X_i-X_j{)}^{-1}]$ for every $w\in D$. Writing $b_0:={\sum }_{w\in D}{T}_w$ and summing, $b_0{u}_{\lambda }\in u_{\mu }{H}_n^{\mathrm{a}\mathrm{f}\mathrm{f}}[(X_i-X_j{)}^{-1}]$.

Finite straightening preserves the left flag. Let $W_{\mu }$ be the span of the $m_{st}$ with $s$ $\mu$-flagged, whether or not $\mu (s)$ is semistandard. Use the finite-generator calculation in step 1.1 of the cellular supplier [F3], rather than only its cell congruence. For right multiplication by $T_i$, a standard interchange retains $s$; a row case is scalar; and a column failure is straightened inside the physical component containing that column. In the last case every left coset word in the ordinary Garnir identity [F4] permutes positions within that component interval, even when its ordinary partition shape changes. Thus before standardization the set of $\mu$-labels in each physical component is unchanged. The increasing component shuffle carries those same sets into the new component diagrams. Quadratic reductions among internal words still permute only those component intervals; replacing a reduced coset word by a Bruhat-smaller word also preserves the flag by step 1.2. Row normalization and further ordinary straightening have the same property. Induction through the finite straightening algorithm therefore retains a flagged left tableau in every term, including higher-shape terms, and proves $W_{\mu }{T}_i\subseteq W_{\mu }$. This argument also proves stability under left $T_i$ for a $\mu$-row generator: its leading interchange exchanges labels of the same type component, preserving the flag, and its nonstandard terms use the internal straightening just analyzed. Consequently $W_{\mu }$ and its intersections with the higher cellular ideals are left modules for the $\mu$-row parabolic.

The exact polynomial-generator correction and its left support. Write $a=a_k$, where $1$ is in component $k$ of $t$, and $d(t)=(s_a\cdots s_1)w$ with $w$ fixing $1$ and length additivity. Put $P=T_1\cdots T_a$, with $P=1$ if $a=0$, and $u_b=u_{\lambda }(X_{a+1}-U_k)$. The defining recurrence and distant commutation give the exact equality $m_{st}{X}_1=z^a{U}_k{T}_{d(s)}^{\ast }{m}_{\lambda }{P}^{-1}{T}_w+z^a{T}_{d(s)}^{\ast }{x}_{\lambda }{u}_b{P}^{-1}{T}_w$, the formula proved in steps 1.2 and 2.1 of [F3]. Its first term is in $W_{\mu }$ by step 2.2, since $T_i^{-1}=z^{-1}(T_i-(z-1))$. For $k=1$, $u_b$ contains the full cyclotomic product in $X_1$ and is zero. For $k>1$, let $\eta$ be the row multicomposition obtained by moving the first node of component $k$ to a new singleton row at the end of component $k-1$. Then $u_{\eta }=u_b$ and $S_{\eta }\subseteq S_{\lambda }$, so the exact length-additive left coset expansion is $x_{\lambda }{u}_b={\sum }_v{T}_v^{\ast }{m}_{\eta }$. Attach the labels of $s$ to their original positions before this expansion: the coset representatives choose which node of the first row of component $k$ occupies the new singleton; all other labels stay in their old physical components. Each selected label had type component at least $k$, so is admissible in component $k-1$ as well. Sorting the rows of $\eta$ and straightening in its physical components preserves the resulting component label sets, as in step 2.2. Since $s$ is row-standard, $d(s)$ is a minimal representative for its left row-subgroup coset, so $T_{d(s)}^{\ast }{T}_v^{\ast }=T_{(vd(s){)}^{-1}}$ is length-additive for these $v\in S_{\lambda }$. Each such reduced coset word has the flagged allocation just described; Bruhat-smaller subwords remain flagged by step 1.2, and internal straightening retains its component label sets. Hence every standard left tableau in $T_{d(s)}^{\ast }{x}_{\lambda }{u}_b$ is flagged. Appending $P^{-1}{T}_w$ preserves this by step 2.2, giving $W_{\mu }{X}_1\subseteq W_{\mu }$. The algebra is generated by $X_1$ and the $T_i$, and the cyclotomic relation expresses $X_1^{-1}$ as a polynomial in $X_1$, since its constant term is a unit. Thus $W_{\mu }$ is a right ideal and $m_{\mu }\in W_{\mu }$ gives $M_{\mu }\subseteq W_{\mu }$. Also $T_{i}m=zm$ for every $m\in M_{\mu }$ and every $\mu$-row generator, because $u_{\mu }$ commutes with that generator and $T_i{x}_{\mu }=z{x}_{\mu }$.

Exact unlocalized membership. Expand $b_0{u}_{\lambda }$ in the left affine PBW basis $X^a{T}_w$ of [F2]; its coefficients lie in $P=R^{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}[X_1^{\pm 1},\dots ,X_n^{\pm 1}]$. Membership in $u_{\mu }{H}_n^{\mathrm{a}\mathrm{f}\mathrm{f}}[(X_i-X_j{)}^{-1}]$ from step 2.1 says each coefficient lies in $u_{\mu }P[(X_i-X_j{)}^{-1}]$. Every factor $X_j-U_k$ of $u_{\mu }$ is a prime of the unique factorization domain $P$, is not associated to any difference $X_i-X_j$, and is not inverted by the localization, so its multiplicity cannot drop when passing to the localization; hence each coefficient is already divisible by $u_{\mu }$ in $P$. Therefore the PBW coordinates of $u_{\mu }^{-1}{b}_0{u}_{\lambda }$ are actual Laurent polynomials and there is an exact unlocalized equality $b_0{u}_{\lambda }=u_{\mu }b$ with $b\in H_n^{\mathrm{a}\mathrm{f}\mathrm{f}}$.

Eigenvalue descent and the explicit element $h$. Let $i$ be a generator of the $\mu$-row subgroup, so that $i,i+1$ lie in one $\mu$-component interval. Expand the double coset into disjoint left cosets, choosing the minimal representative $r$ of each coset. Length additivity gives $b_0={\sum }_r{x}_{\mu }{T}_r$, so $T_i{b}_0=z{b}_0$ by pairing the row-subgroup words under left multiplication by $s_i$; right multiplication by $u_{\lambda }$ retains this left eigenvalue, so $T_i(u_{\mu }b)=z{u}_{\mu }b$. The element $u_{\mu }$ is symmetric in the variables of each $\mu$-interval, so it commutes with $T_i$, and injectivity of left multiplication by the nonzero Laurent polynomial $u_{\mu }$ ([F2]) cancels it, giving $T_{i}b=zb$. Write $b={\sum }_w{T}_w{f}_w$ in the right PBW basis ([F2]). For each $w$ with $\mathrm{\ell }(s_{i}w)=\mathrm{\ell }(w)+1$, comparing coefficients of $T_{s_{i}w}$ in $T_{i}b=zb$ gives $f_w+(z-1)f_{s_{i}w}=z{f}_{s_{i}w}$ by the quadratic relation $(T_i-z)(T_i+1)=0$, hence $f_w=f_{s_{i}w}$; thus $f$ is constant on each left coset $S_{\mu }w$. Taking minimal-length coset representatives $r$ and using the length-additive expansion $x_{\mu }{T}_r={\sum }_{\sigma \in S_{\mu }}{T}_{\sigma r}$ gives $b=x_{\mu }{h}_0$ with $h_0={\sum }_r{T}_r{f}_r\in H_n^{\mathrm{a}\mathrm{f}\mathrm{f}}$. Since $x_{\mu }$ commutes with $u_{\mu }$, the explicit sum defining $m_{(S,t)}$ becomes $m_{(S,t)}=b_0{u}_{\lambda }{T}_{d(t)}=u_{\mu }{x}_{\mu }{h}_0{T}_{d(t)}=m_{\mu }h$ with $h=h_0{T}_{d(t)}$; this constructs $h$ by finite coset enumeration and polynomial division with no inverted quantum factorial, and the equality is exact, so the Murphy coefficient at the pivot $m_{s_{0}t}$ is $1$.

The flagged multiplicity upper bound. All finite row parabolics are split semisimple over $K$ by [F4]. In an allocation summand of step 1.3, the multiplicity of their trivial character equals the multiplicity of the trivial intersection character on ${⨂}_b{V}_b$. This can be seen directly: the finite parabolic symmetrizer divided by its nonzero generic quantum factorial is the trivial-character projector, and its coset expansion identifies the invariant vectors of the induced module with the invariant vectors for the intersection. For each component $b$ put ${\rho }_b=(N_{r,b}{)}_r$, omitting zero parts. By [F4] this intersection multiplicity is ${\prod }_b{K}_{({\lambda }^{(b)},{\rho }_b)}$. Summing only over the flag-compatible allocations therefore gives $\mathrm{\#}{\mathrm{SStd}}_{\mu }(\lambda )$: choose an ordinary semistandard tableau in each component with the prescribed row-label multiplicities and combine them; weak-row and strict-column conditions are componentwise and the allocation supplies exactly the component flag. The cellular-ideal filtration of $W_{\mu }$ from steps 2.2 and 3.1 has in each shape the flagged left cell tensored with the full right cell, so exactness of the trivial-character projector gives ${\dim }_K(W_{\mu }{)}^{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{v}}={\sum }_{\lambda }\mathrm{\#}{\mathrm{SStd}}_{\mu }(\lambda )\mathrm{\#}{\mathrm{Std}}_{\lambda }$. Step 3.1 puts $M_{\mu }^K$ inside this space; step 4.1 puts all that many independent fibre sums inside $M_{\mu }^K$. The two bounds agree, proving that $M_{\mu }^K$ is their $K$-span. The generic projector is used only for this dimension bound; the membership identities and subsequent descent divide by no quantum factorial.

Descent to $R$, specialization, direct summand, and Specht filtration. Let $h\in m_{\mu }{H}_R$. Its Murphy coordinates in the basis [F3] lie in $R$, and step 5.1 says that, after extending scalars to $K$, they vanish outside the admissible fibres and are constant on each admissible fibre; these are literal coefficient identities in $R$ because the Laurent polynomial ring injects into $K$, so $h$ lies in the span $V_{\mu }$ of the $m_{(S,t)}$. Hence $M_{\mu }\subseteq V_{\mu }$, and step 4.1 gives the reverse inclusion, so $M_{\mu }=V_{\mu }$ over $R$; every element of $V_{\mu }$ was constructed as $m_{\mu }h$, so this infers no basis from generic flatness. For any commutative specialization $R^{\mathrm{\prime }}$, the same two inclusions hold after applying $-{\otimes }_R{R}^{\mathrm{\prime }}$: each $m_{(S,t)}$ remains in $m_{\mu }{H}_{R^{\mathrm{\prime }}}$ by the explicit identity of step 4.1, and each element of $m_{\mu }{H}_{R^{\mathrm{\prime }}}$ has Murphy coordinates in $V_{\mu }$ by the specialized coefficient identities. Finally, with $s_0$ the pivot refinement of $S$, replace in the Murphy basis the vector $m_{s_{0}t}$ by the fibre sum $m_{(S,t)}$ and retain $m_{st}$ for all non-pivot $(s,t)$: this change of basis is unitriangular with diagonal entries $1$ over $R$, so the $m_{(S,t)}$ have disjoint coefficient-$1$ pivots and span an $R$-direct summand of $H_n$. Ordering the semistandard $S$ in a shape-compatible order, the Murphy cell law [F3] identifies each successive fixed-fibre quotient with the Specht module $S_{\lambda }$, one copy for each semistandard $S$.