Modular Branching and the Good-Node Crystal

An integral symmetric polynomial evaluates in $\mathbb{Z}{S}_n$. Its commutator with any permutation vanishes in $\mathbb{C}{S}_n$ by [F1], hence vanishes integrally by coefficientwise injectivity. For any coefficient ring, invariance of a polynomial under variable permutations says exactly that its coefficients are constant on monomial orbits, so it is a linear combination of integral monomial symmetric polynomials. All their evaluations are therefore central over $R$. A group-ring element is central exactly when its coefficients are constant on conjugacy classes: compare coefficients after conjugating by each permutation. Consequently the class sums form a basis of the center over $\mathbb{Z}$ and over $R$, even with zero divisors.

Define the reduced cycle type $\rho (g)$ by subtracting one from each nontrivial cycle length and discarding fixed points. Its size $d(g)=n-c(g)$ is the minimum number of transpositions expressing $g$: multiplying by a transposition changes the number of cycles by one, giving the lower bound, and each cycle of length $s$ has a factorization with $s-1$ transpositions, giving equality. The possible types are precisely partitions $\rho$ with $\mathrm{\mid }\rho \mathrm{\mid }+\mathrm{\ell }(\rho )\le n$. Write $K_{\rho }$ for their integral class sums.

If a word of $d$ transpositions has product of degree $d$, every prefix multiplication joins two different cycles; otherwise the final degree would be at most $d-2$. The graph whose edges are the transpositions is therefore a forest, and its connected components are exactly the cycles of every prefix product. Indeed initially both are singletons, and each multiplication joins two current components by one edge and joins their two cycles. Thus edges never repeat, and each final tree of $s$ vertices gives a cycle of length $s$.

Let $\rho$ be a possible reduced type of size $d$, and expand the monomial symmetric polynomial $m_{\rho }(L_1,\dots ,L_n)$, summing once over distinct assignments of the positive parts of $\rho$ to distinct indices $r$. Each term is a word of $d$ transpositions, grouped into stars centered at those $r$: the group $L_r^a$ has $a$ edges incident to $r$. Every resulting permutation has degree at most $d$. If its degree is $d$, step 2.1 makes all these stars trees with distinct edges. Each star lies in one final tree; the degree of that tree is the sum of the exponents of its stars. Its reduced cycle type is therefore obtained from $\rho$ by merging some parts. Any proper merging decreases the number of parts.

The coefficient at any permutation $g$ of reduced type exactly $\rho$ is one. In the degree-$d$ expansion, equality of the number of parts forces each final tree to contain exactly one star. For a cycle with support $\{r,a_1,\dots ,a_s\}$, the center of its star must be its largest letter $r$, since every factor from $L_r$ is $(a,r)$ with $a<r$. The ordered star factorization is unique: with right-to-left permutation composition, $(a_1,r)\cdots (a_s,r)$ is the cycle $(r,a_s,\dots ,a_1)$, as direct evaluation on each letter shows. Thus the cycle determines the leaf ordering. Different cycles have distinct maxima, their stars are disjoint and commute, and their assignments appear exactly once among the distinct monomials, including repeated exponents. This also constructs a term producing $g$.

By centrality and steps 3.1–4.1, $m_{\rho }(L)=K_{\rho }+{\sum }_{\sigma }{a}_{\rho \sigma }{K}_{\sigma }$, with integer coefficients, where every remaining type either has smaller degree or is a proper merging of $\rho$. Induct on degree and, within a degree, on the number of parts in increasing order. All remaining sums are already integral symmetric polynomial evaluations, so subtraction expresses $K_{\rho }$ that way too. The initial type is empty with $m_{\mathrm{\varnothing }}=K_{\mathrm{\varnothing }}=1$. This proves integral spanning without division. Specialization to $R$ and the class-sum basis from step 1.1 prove the assertion over every $R$, including $n=0$ and the zero ring.

Eigenvalues in ${\mathbb{F}}_p$. Let $h_r\in \mathbb{Q}[X]$ be the minimal polynomial of $L_r$ acting on the left regular module $\mathbb{Q}{S}_n$. Over $\mathbb{C}$ the regular module is the direct sum of the complex Specht modules with multiplicities, and by [F2] the eigenvalues of $L_r$ on that module are the integers $c_T(r)$, the contents of the node containing $r$, over all standard tableaux $T$ of $n$ nodes; hence $h_r={\prod }_{a\in E_r}(X-a)$ is squarefree with the integer roots in the finite set $E_r$. The group-basis coordinates of $h_r(L_r)$ are integers and vanish over $\mathbb{Q}$, so this identity holds already in $\mathbb{Z}{S}_n$. Reduction of coefficients modulo $p$ therefore gives $h_r(L_r)=0$ in $k{S}_n$, so the minimal polynomial $f_r$ of $L_r$ on the regular module $k{S}_n$ divides the reduction ${\bar{h}}_r={\prod }_{a\in E_r}(X-\bar{a})$ in $k[X]$, and every root of $f_r$ lies in the image ${\mathbb{F}}_p$ of $\mathbb{Z}$ in $k$.

Antipode invariance. The antipode $\sigma :k{S}_n\to k{S}_n$, $\sigma (g)=g^{-1}$, is an anti-automorphism fixing every transposition, hence fixes each $L_r$ and therefore each $P_{r,a}\in k[L_r]$ and each $e(\mathbf{i})$.

An integral central identity. In $\mathbb{Z}{S}_n$ one has ${\prod }_{r=1}^n(u+L_r)={\sum }_{g\in S_n}{u}^{c(g)}g$, where $c(g)$ is the number of cycles, including fixed points. To prove this by induction, regard $S_{n-1}$ as fixing $n$. A term multiplied by $u$ leaves $n$ fixed and adds one cycle. A term multiplied by $(a,n)$ inserts $n$ in the cycle containing $a$, without changing its cycle count. Conversely, deleting a nonfixed $n$ from its cycle uniquely recovers the permutation in $S_{n-1}$ and the transposition used. Thus every permutation occurs exactly once. Comparing coefficients, each elementary symmetric polynomial $C_s$ in $L_1,\dots ,L_n$ is a sum of all permutations with $n-s$ cycles and hence is central; the identity and centrality remain true after reduction to $k$.

Primary projectors. Write $f_r={\prod }_{a\in {\mathbb{F}}_p}(X-a{)}^{m_a}$ with $m_a\ge 0$. For $a\mathrm{\ne }b$ the ideals $((X-a{)}^{m_a})$ and $((X-b{)}^{m_b})$ of $k[X]$ are comaximal, since two powers of distinct linear polynomials are coprime and Bezout gives $1=u(X)(X-a{)}^{m_a}+v(X)(X-b{)}^{m_b}$. By [F4] applied to the ring $k[X]\mathrm{/}(f_r)$ and its comaximal ideals, the canonical map $k[X]\mathrm{/}(f_r)\to {\prod }_{a:m_a>0}k[X]\mathrm{/}((X-a{)}^{m_a})$ is an isomorphism; let ${\epsilon }_{r,a}$ be the preimage of the idempotent supported at the $a$-th factor when $m_a>0$, and put ${\epsilon }_{r,a}=0$ when $m_a=0$. Then ${\sum }_a{\epsilon }_{r,a}=1$, ${\epsilon }_{r,a}^2={\epsilon }_{r,a}$, ${\epsilon }_{r,a}{\epsilon }_{r,b}=0$ for $a\mathrm{\ne }b$, and $(X-a{)}^{m_a}{\epsilon }_{r,a}=0$ in $k[X]\mathrm{/}(f_r)$. Substituting the algebra element $L_r$ for $X$ (legitimate because $f_r(L_r)=0$) gives elements $P_{r,a}:={\epsilon }_{r,a}(L_r)\in k[L_r]$ with ${\sum }_a{P}_{r,a}=1$, $P_{r,a}{P}_{r,b}={\delta }_{a,b}{P}_{r,a}$ and $(L_r-a{)}^{m_a}{P}_{r,a}=0$.

These are the primary projections. Let $M$ be a finite-dimensional $k{S}_n$-module. Since the $P_{r,a}$ are polynomials in $L_r$, they commute with $L_r$ and with one another; as $1={\sum }_a{P}_{r,a}$ one has $M={\sum }_a{P}_{r,a}M$, and $(L_r-a{)}^{m_a}$ annihilates $P_{r,a}M$, so $P_{r,a}M\subseteq M_{r,a}$. Conversely, if $m\in M_{r,a}$, then $(L_r-a{)}^{N}m=0$ for $N\ge m_a$, and for $b\mathrm{\ne }a$ the element $L_r-b=(a-b)+(L_r-a)$ is invertible on the $a$-primary subspace because $(a-b)\in k^{\times }$ and $(L_r-a)$ is nilpotent there; from $P_{r,b}(L_r-b{)}^{m_b}=0$ we get $P_{r,b}m=0$ for $b\mathrm{\ne }a$, whence $m=P_{r,a}m$. Thus $P_{r,a}M=M_{r,a}$ and $M={⨁}_a{M}_{r,a}$. The products $e(\mathbf{i})={\prod }_r{P}_{r,i_r}$ are idempotent, pairwise orthogonal and sum to $1$ by multiplicativity of the $P_{r,a}$; applying the same argument successively to the commuting operators $L_1,\dots ,L_n$ gives $e(\mathbf{i})M=M_{\mathbf{i}}$.

Central CRT separates contents. Choose $N\ge 1$ such that $(L_r-i_r{)}^{N}e(\mathbf{i})=0$ for every $r$ and every tuple $\mathbf{i}$, using step 2.1 and the finiteness of the indices. On the commutative algebra summand $e(\mathbf{i})k[L_1,\dots ,L_n]$ with identity $e(\mathbf{i})$, the commuting elements $(L_r-i_r)e(\mathbf{i})$ generate an ideal whose $D$-th power is zero for $D=n(N-1)+1$: every monomial of degree $D$ has a factor with exponent at least $N$. Thus $C_s-c_s(\mathbf{i})$ belongs to that nilpotent ideal, where $c_s(\mathbf{i})$ is the elementary symmetric function of the residues. The tuple $c(\mathbf{i})=(c_1(\mathbf{i}),\dots ,c_n(\mathbf{i}))$ determines the content, because ${\prod }_r(X-i_r)={\sum }_{s=0}^n(-1{)}^s{c}_s(\mathbf{i})X^{n-s}$ determines the multiplicities of its distinct linear factors, over every characteristic. In $R=k[Y_1,\dots ,Y_n]$, the distinct content tuples therefore define distinct pairwise comaximal maximal ideals ${\mathfrak{m}}_{\alpha }=(Y_s-c_s(\alpha ):1\le s\le n)$; their powers ${\mathfrak{m}}_{\alpha }^D$ are also pairwise comaximal, since expanding $1=(a+b{)}^{2D-1}$ for $a+b=1$ places every term in one of the two $D$-th powers. By [F4], choose $Q_{\alpha }\in R$ congruent to $1$ modulo ${\mathfrak{m}}_{\alpha }^D$ and to $0$ modulo every other ${\mathfrak{m}}_{\beta }^D$. The nilpotence just proved gives $Q_{\alpha }(C_1,\dots ,C_n)e(\mathbf{i})=e(\mathbf{i})$ when $\mathbf{i}\in I_{\alpha }$, and $0$ otherwise. Since the tuple idempotents sum to $1$, it follows that $Q_{\alpha }(C_1,\dots ,C_n)={\sum }_{\mathbf{i}\in I_{\alpha }}e(\mathbf{i})=e_{\alpha }$. This is central by step 1.3. Idempotency, pairwise orthogonality for distinct content classes, and ${\sum }_{\alpha \in {\mathcal{C}}_n}{e}_{\alpha }=1$ follow from step 3.1 and the partition of all tuples into their content classes.

The functor projectors. Fix $i\in {\mathbb{F}}_p$ and $\alpha \in {\mathcal{C}}_{n-1}$. The primary projectors for $L_r$, $r<n$, computed in $k{S}_{n-1}$ agree with those in $k{S}_n$: the primary idempotents in $k[L_r]$ are uniquely determined by their congruences modulo the powers of the distinct linear factors of the minimal polynomial, and the inclusion preserves the polynomial relations of $L_r$. Summing once over each distinct sequence representing $\alpha$ gives $e_{\alpha ,{\alpha }_i}={\sum }_{\mathbf{j}\in I_{\alpha }}e((\mathbf{j},i))=e_{\alpha }^{(n-1)}{P}_{n,i}$. By step 4.1 the first factor is central in $k{S}_{n-1}$, while the second commutes with $k{S}_{n-1}$ by [F1]; their product is therefore an idempotent commuting with that subgroup algebra. The sets $\{(\mathbf{j},i):\mathbf{j}\in I_{\alpha }\}$ are disjoint for distinct pairs $(\alpha ,i)$ and partition all tuples of length $n$. Consequently these idempotents are pairwise orthogonal and sum to $1$, and their images are $k{S}_{n-1}$-submodules. Grouping these actual direct summands by the last residue gives $\mathrm{Res}M={⨁}_{i\in {\mathbb{F}}_p}{E}_{i}M$. The indexing is by content classes throughout, so no content is counted twice.

Every nonzero content summand has local center. Fix $e=e_{\alpha }\mathrm{\ne }0$. Since $e$ is central, $Z(ek{S}_n)=eZ(k{S}_n)$: an element central in the summand commutes with the complementary summand because their products vanish. By [F7] every element of this center is $ef(L)$ for a symmetric polynomial $f$ over $k$. For every $\mathbf{i}\in I_{\alpha }$, the difference $(f(L)-f(\mathbf{i}))e(\mathbf{i})$ belongs to the nilpotent ideal of step 4.1; symmetry makes $f(\mathbf{i})$ the same scalar $a=f(\alpha )$ for all these tuples. The orthogonal sum of these differences is nilpotent with the common exponent $D$ of that step. Hence every central element of the summand has the form $ae+u$ with $u$ nilpotent. The scalar $a$ is unique: a nonzero scalar multiple of $e$ is invertible in this nonzero summand and cannot be nilpotent. As the center is commutative, sums of commuting nilpotents and their products with any central element are nilpotent. Thus its nilpotents form an ideal, the scalar map is a surjective ring homomorphism onto $k$, and every element with nonzero scalar is invertible by the finite geometric series. The center is local over the original field $k$, without extending scalars.

Induction side. The same computation with $n$ replaced by $d+1$ shows that $e_{\alpha ,{\alpha }_i}=e_{\alpha }^{(d)}{P}_{d+1,i}$ is an idempotent of $k{S}_{d+1}$ commuting with $k{S}_d$, that ${\sum }_{i,\alpha }{e}_{\alpha ,{\alpha }_i}=1$ with pairwise orthogonal summands, and that $k{S}_{d+1}={⨁}_{i,\alpha }k{S}_{d+1}{e}_{\alpha ,{\alpha }_i}$; hence for any finite-dimensional $M$, $k{S}_{d+1}{\otimes }_{k{S}_d}M={⨁}_{i,\alpha }k{S}_{d+1}{e}_{\alpha ,{\alpha }_i}{\otimes }_{k{S}_d}{e}_{\alpha }M={⨁}_i{F}_{i}M,$ the middle step using $M={⨁}_{\alpha }{e}_{\alpha }M$ and the centrality of $e_{\alpha }$ in $k{S}_d$. This is $\mathrm{Ind}M={⨁}_i{F}_{i}M$.

Primitivity and exhaustiveness. A central idempotent $b$ in $ek{S}_n$ has scalar $0$ or $1$ by step 5.2. In the first case $b$ is nilpotent and idempotent, so $b=0$; in the second case $e-b$ is nilpotent and idempotent, so $b=e$. Thus each nonzero $e_{\alpha }$ is primitive central. Their orthogonal sum is $1$ by step 4.1. Any primitive central idempotent $b$ of $k{S}_n$ decomposes as ${\sum }_{\alpha }b{e}_{\alpha }$ into orthogonal central idempotents; primitivity leaves exactly one nonzero term, and the primitivity of that $e_{\alpha }$ forces $b=e_{\alpha }$. By [F6] these are precisely the block idempotents. At rank zero $k{S}_0=k$ and $e(\mathrm{\varnothing })=1$ gives the unique block; restriction from rank zero is taken to be the zero functor, while induction to rank one has only residue $0$. This convention does not invoke a group $S_{-1}$.

Exactness and duality. Each $E_i$ is a direct sum of the functors $M\mapsto e_{\alpha ,{\alpha }_i}M$; a map $f:M\to N$ of $k{S}_n$-modules commutes with the action of the element $e_{\alpha ,{\alpha }_i}$, so it restricts to a map $e_{\alpha ,{\alpha }_i}M\to e_{\alpha ,{\alpha }_i}N$, and each such functor is exact as a direct summand of the identity, hence additive and finite-dimensionality preserving. Each $F_i$ is a direct sum of the functors $M\mapsto k{S}_{d+1}{e}_{\alpha ,{\alpha }_i}{\otimes }_{k{S}_d}{e}_{\alpha }M$, which are direct summands of induction $M\mapsto k{S}_{d+1}{\otimes }_{k{S}_d}M$; by [F5] $k{S}_{d+1}$ is free as a right $k{S}_d$-module, so induction, and with it each $F_i$, is exact, additive and finite-dimensionality preserving. For duality observe that $e_{\alpha ,{\alpha }_i}$ is fixed by the antipode by step 1.2, and that the standard nondegenerate pairing $M\times M^{\ast }\to k$ satisfies $(xm,f)=(m,x^{\ast }f)$; hence for finite-dimensional $M$ the transpose of the projection on $M$ is the projection on $M^{\ast }$ and $(e_{\alpha ,{\alpha }_i}M{)}^{\ast }\cong e_{\alpha ,{\alpha }_i}{M}^{\ast }$, giving $E_i(M{)}^{\ast }\cong E_i(M^{\ast })$. All these restriction-duality identifications are $k$-linear and natural.

Induction duality by a perfect coset pairing. Put $G=S_{d+1}$, $H=S_d$, and let ${\pi }_H:kG\to kH$ retain the coefficients of the elements of $H$. For a finite-dimensional left $kH$-module $M$, with contragredient action $(af)(m)=f(a^{-1}m)$, define $⟨g\otimes m,h\otimes f⟩=f({\pi }_H(h^{-1}g)m)$ between $\mathrm{Ind}M$ and $\mathrm{Ind}{M}^{\ast }$. For $a\in H$, the $kH$-bimodule property of ${\pi }_H$ gives ${\pi }_H(h^{-1}ga)={\pi }_H(h^{-1}g)a$ and ${\pi }_H((ha{)}^{-1}g)=a^{-1}{\pi }_H(h^{-1}g)$; these identities prove balancing in the first and second tensor factors, respectively. Simultaneous left multiplication of $g,h$ by any element of $G$ leaves $h^{-1}g$ unchanged, proving $G$-invariance. On a finite left-coset transversal $T$ one has ${\pi }_H(s^{-1}r)={\delta }_{s,r}1$ for $r,s\in T$. Thus the pairing is the direct sum of the perfect evaluations $M\times M^{\ast }\to k$ and is perfect by [F5]. For $e=e_{\alpha ,{\alpha }_i}$, right multiplication $R_e(g\otimes m)=ge\otimes m$ is a well-defined $G$-linear idempotent because $e$ commutes with $kH$. Let $\tau :kG\to k$ be the coefficient of $1$; directly on group elements, $\tau (xy)=\tau (yx)$. For every $b\in H$, the coefficient of $b$ in $ze$ equals $\tau (b^{-1}ze)=\tau (e{b}^{-1}z)=\tau (b^{-1}ez)$, since $e$ commutes with $b$. Therefore ${\pi }_H(ze)={\pi }_H(ez)$ for all $z\in kG$. Writing $e^{\ast }$ for its antipode image, step 1.2 gives $e^{\ast }=e$, so expansion in the group basis yields $⟨R_e(g\otimes m),h\otimes f⟩=f({\pi }_H(h^{-1}ge)m)=f({\pi }_H(e{h}^{-1}g)m)=⟨g\otimes m,R_e(h\otimes f)⟩$. The two images of $R_e$ pair perfectly: a vector in either image annihilating the other annihilates the entire opposite space, since the opposite kernel is orthogonal to that image, and hence vanishes by perfectness. By tensor balancing, $ge\otimes m=ge\otimes e_{\alpha }m$, so these images are precisely $kGe{\otimes }_{kH}{e}_{\alpha }M$ and $kGe{\otimes }_{kH}{e}_{\alpha }{M}^{\ast }$. Distinct $e_{\alpha ,{\alpha }_i}$ images are orthogonal by self-adjointness and projector orthogonality. Summing the perfect image pairings over $\alpha \in {\mathcal{C}}_d$ proves the natural $G$-isomorphism $F_i(M^{\ast })\cong F_i(M{)}^{\ast }$; invariance supplies the contragredient action, and evaluation shows naturality in $M$. The construction uses finite transversals and holds over every characteristic, including $2$.

Partition content agrees with the block content. Let $\lambda ⊢n$ and let $\alpha$ be the multiset of its node contents reduced modulo $p$. Over $\mathbb{C}$, [F2] makes every elementary symmetric $C_s$ act on $S^{\lambda }$ by the integer $c_s(\lambda )$. In the integral basis of [F8] the matrix of $C_s-c_s(\lambda )$ has integer entries and vanishes after tensoring with $\mathbb{C}$, so it vanishes integrally and after specialization to $k$. Thus the CRT polynomial $Q_{\alpha }(C_1,\dots ,C_n)$ of step 4.1 acts as $1$ on $S_k^{\lambda }$, while the other content projections act as zero. The same is true on its nonzero subquotients. Hence any simple module obtained from this Specht module belongs to the block $e_{\alpha }$, establishing the stated content dictionary without a splitting-field descent or a reduced interpolation denominator.