Kleshchev's first-level branching multiplicity

Use the actual $q=1$ Schur Weyl-cell simple $L_N(\mu )$, $\mu$ dominant polynomial, converted from right to left by star. Its highest vector $v$ has weight $\mu$ and norm $1$. The local lemma [F3] proves from the tensorword endomorphism realization that negative coefficient operators span each weight space, positive coefficient operators detect every nonzero vector, and integral cut-root words span every grade whose simple-root coefficients are $0$ or $1$. It also proves the complete essential-subset basis and the dimension formula $\dim L_N(\mu {)}_{\mu -\alpha (r,s)}={\epsilon }_a(r,s)$, $a_j={\mu }_j-{\mu }_{j+1}$. The local lemma [F3] also proves for every dominant $a$ and every interval, by finite matching/path/stack bijections and finite triangular matrix inversion, ${\epsilon }_a(r,s)={\nu }_a(r,s)+{\sum }_{u=r+1}^s{\epsilon }_a^{\mathrm{\prime }}(r,u-1){\nu }_a(u,s)$, where ${\epsilon }_a^{\mathrm{\prime }}(r,t)={\epsilon }_{a+{\omega }_t}(r,t)$. These are actual proved local inputs, including all zero dimensions and arbitrary nonrestricted auxiliary weights.

We spell out a further modular consequence of the same foundation to close the algebraic support seam. Let $\mathrm{\Gamma }$ be a deficit with each simple-root coefficient $0$ or $1$, with disconnected support $\mathrm{\Gamma }=\delta +\zeta$ separated by at least one zero coefficient. Take the first component $\delta$ entirely inside the Levi on the first $N-1$ coordinates. Every negative cut word of grade $\mathrm{\Gamma }$ factors uniquely into a $\delta$-word and a $\zeta$-word, since no root crossing a missing simple root can occur; the two groups commute. Every positive coefficient word of grade $\mathrm{\Gamma }$ similarly factors. All positive/negative cross commutators between the separated groups vanish. Consequently the matrix of highest coefficients at $\mathrm{\Gamma }$ is the tensor product of the two separate coefficient matrices at $\delta$ and $\zeta$: for positive tests $e,e^{\mathrm{\prime }}$ and negative words $f,f^{\mathrm{\prime }}$, $\mathrm{highest}(e{e}^{\mathrm{\prime }}f{f}^{\mathrm{\prime }}v)=\mathrm{highest}(efv)\mathrm{highest}(e^{\mathrm{\prime }}{f}^{\mathrm{\prime }}v)$. The rightmost separated test/word leaves the Cartan coordinates of the other interval unchanged, since the intervals are nonadjacent; this verifies the displayed equality even though $f^{\mathrm{\prime }}fv$ has shifted weight.

Choose bases $f_{a}v$ and $f_b^{\mathrm{\prime }}v$ of the separate images. Positive detection yields positive linear-combination tests $e_a$ and $e_b^{\mathrm{\prime }}$ with $e_a{f}_{c}v={\delta }_{ac}v$, $e_b^{\mathrm{\prime }}{f}_d^{\mathrm{\prime }}v={\delta }_{bd}v$, by finite matrix inversion over the field. Thus the products $f_a{f}_b^{\mathrm{\prime }}v$ are independent, since $e_a{e}_b^{\mathrm{\prime }}$ tests detect their coefficients; they span by the preceding factorization. On the entire weight space at $\mathrm{\Gamma }$ the operator $P_{\delta }={\sum }_a{f}_a{e}_a$ is the identity: applied to $f_c{f}_b^{\mathrm{\prime }}v$, its cross commutations give $f_c{f}_b^{\mathrm{\prime }}v$. Every $e_a$ is a positive Levi coefficient word of nonzero weight $\delta$, and every $f_a$ is a negative Levi coefficient word. This argument proves the identity on arbitrary entangled linear combinations, not only pure products. If a separate image is zero, the full weight space is zero and cannot supply a primitive vector.

Now let $M_0\subset M_1$ be consecutive terms of a finite composition series of the first level, and choose $x\in M_1$ representing the highest vector of a simple factor, of full deficit $\mathrm{\Gamma }$. Positive Levi operators of nonzero weight send $x$ into $M_0$, because they annihilate its highest class. If $\mathrm{\Gamma }$ had a disconnected initial component $\delta$, the identity $P_{\delta }x=x$ would put $x$ into $M_0$: $e_{a}x\in M_0$ and $f_a$ preserve $M_0$. This is impossible. Hence any primitive factor vector of binary deficit ending at the last simple root has support a single terminal interval. This is the disconnected-root primitive exclusion, proved here, without a characteristic-zero PBW theorem or a pure-product assumption for $x$.

Suppose ${\mu }_N=0$ and let $l=N-1$. Define the first level $M$ as the sum of weight spaces of $L_N(\mu )$ whose last diagonal coordinate is $1$. It is a polynomial Schur/${\mathrm{G}\mathrm{L}}_{N-1}$ module of degree $\mathrm{\mid }\mu \mathrm{\mid }-1$; the Levi preserves that coordinate. Put ${\beta }_i={\alpha }_i+\cdots +{\alpha }_l$. For any $i$ write ${\nu }_i=({\mu }_1,\dots ,{\mu }_{N-1})-{ϵ}_i$, as a weight, and write $d_i=\dim L_N(\mu {)}_{\mu -{\beta }_i}$, $b_u=[M:L_{N-1}({\nu }_u)]$ when ${\nu }_u$ is dominant, and $b_u=0$ otherwise. Dominance of ${\nu }_u$ is equivalent to $a_u>0$. No modular restriction condition is imposed on a dominant ${\nu }_u$. In particular non-$p$-restricted factors still contribute to all character equations below.

A simple factor contributing to the ${\nu }_i$ weight must have highest weight $\eta$ dominating ${\nu }_i$. The standard Schur cell simples have unique highest vector, all other weights dominated by it, and cover all simple Schur modules by the cellular regular-module argument. Thus this statement applies to every factor over every field. Lift $\eta$ by appending the last coordinate $1$, and let $\mathrm{\Gamma }=\mu -(\eta ,1)$. Since $\eta$ is a weight of the first level, $\mathrm{\Gamma }$ is a nonnegative integral combination of simple roots; since $\eta$ dominates ${\nu }_i$, ${\beta }_i-\mathrm{\Gamma }$ is a nonnegative integral combination of Levi roots. Consequently ${\mathrm{\Gamma }}_j=0$ for $j<i$, ${\mathrm{\Gamma }}_j\in \{0,1\}$ for $i\le j<l$, and ${\mathrm{\Gamma }}_l=1$. The support exclusion just proved forces $\mathrm{\Gamma }={\beta }_u$ for some $u\ge i$. Thus $\eta ={\nu }_u$ and ${\nu }_u$ must be dominant. Only these terminal labels can contribute to this character equation; no stronger assertion about the other first-level factors is needed.

Weight-space dimensions are additive on a composition series. Indeed the actual weight idempotents give direct projections, so taking a fixed torus weight is exact. The own highest weight of $L_{N-1}({\nu }_i)$, when defined, has dimension $1$. Thus $d_i=b_i+{\sum }_{u>i}{b}_u{a}_{iu}$, $a_{iu}=\dim L_{N-1}({\nu }_u{)}_{{\nu }_i}$. For an invalid ${\nu }_u$ its term is omitted. The lower weight deficit is ${\nu }_u-{\nu }_i={ϵ}_i-{ϵ}_u={\alpha }_i+\cdots +{\alpha }_{u-1}$. The fundamental coordinates of ${\nu }_u$ on that interval are $a_i,\dots ,a_{u-2},a_{u-1}+1$. Any altered coordinate at $u$ is outside the interval and has no effect on the local formulas. The proved interval-basis theorem therefore gives $d_i={\epsilon }_a(i,l)$, $a_{iu}={\epsilon }_a^{\mathrm{\prime }}(i,u-1)$. This is valid for every legitimate dominant ${\nu }_u$, including those not $p$-restricted. When $a_u=0$, ${\nu }_u$ is nondominant and $b_u=0$; also ${\nu }_a(u,l)=0$ because its root is not normal.

Substitute these dimensions into the character equation and compare it with the local recurrence of [F3] at $(i,l)$: ${\epsilon }_a(i,l)=b_i+{\sum }_{u>i}{b}_u{\epsilon }_a^{\mathrm{\prime }}(i,u-1)$, ${\epsilon }_a(i,l)={\nu }_a(i,l)+{\sum }_{u>i}{\nu }_a(u,l){\epsilon }_a^{\mathrm{\prime }}(i,u-1)$. Starting at $i=l$ and proceeding downwards proves $b_i={\nu }_a(i,l)$ for all $i$. This is finite integer reverse substitution, never reduction of the character multiplicities modulo $p$. The rank is $N-1$, unrelated to the prime except in the $B,C$ zero tests.

The local lemma [F3] uses only the infinite purely transcendental extension $K=F(t)$ for polynomial identities and Gaussian elimination. Its scalar-extension argument is reversible: all cut/raising matrices have entries in $F$, hence their finite ranks and kernels preserve extension; its essential labels and pivot tests yield bases over $F$ itself. Schur cell simples remain simple under every field extension by the cellular rank-one action and their nonzero form. Extending a finite Schur composition series thus retains the simple factors and their labels, so the integers $b_i$ computed over $K$ are exactly those over $F$. The first-level weight subspace commutes with extension because it is the image of a finite sum of weight idempotents. Neither an algebraic closure nor an assumption that the original field is infinite is used. All nil/zero-weight cases remain in the equations.

Let $\lambda$ be $p$-regular, $\lambda \mathrm{\ne }(n)$, put $N=n$, $\mu ={\lambda }^{\mathrm{\prime }}$, ${\mu }_N=0$. The index of $A=(j,{\lambda }_j)$ is $i={\lambda }_j$, not $j$, and not $p$. The coordinate $a_u={\mu }_u-{\mu }_{u+1}$ is the multiplicity of the part $u$ in $\lambda$; in particular $a_u<p$, but the preceding algebraic proof did not need that restriction. A removable $A$ is the last row of its width $i$ block, hence $j={\mu }_i$.

The root normal walk based at $i$ starts with a push for $A$ and scans $u=i+1,\dots ,l$. At coordinates with $a_u=0$ the compensation walk is neutral, even when $B=C=0$: the same position supplies and consumes a compensation slot. At positive coordinates $u$, the last ordinary row $k$ of the width $u$ block gives a removable node, and $C(i,u)=u-i+{\sum }_{t=i}^{u-1}{a}_t=u-{\lambda }_j+j-k$. Thus $C(i,u)=0$ is exactly the removable node's residue equal to $A$'s residue. The first ordinary row $k$ of the width $u$ block has an addable node in column $u+1$, and $B(i,u)=u-i+{\sum }_{t=i}^u{a}_t=u-{\lambda }_j+j-k+1$. Thus $B(i,u)=0$ is exactly that addable node's residue equal to $A$'s residue. Since $0<a_u<p$ at a positive coordinate, $B=C+a_u$ shows these two events cannot occur together at that same $u$. Empty widths give only the neutral compensation case. The artificial final zero-width row also has no effect. Scanning positive $u$ in increasing order lists the ordinary corners above $A$ in the order $j-1,\dots ,1$, with a width block contributing its boundary event when the residue matches. The top extra addable node of width ${\lambda }_1+1$ is already the $B$ event of the width ${\lambda }_1$ coordinate; no boundary event is omitted.

Under transpose a removable ordinary node becomes a removable $\mu$-node of opposite residue, and an addable becomes an addable of opposite residue. The top-to-bottom $\mu$-signature following $A^{\mathrm{\prime }}$ lists exactly these above ordinary nodes in this order. The only possible equal-column lower ordinary addable node lies immediately below $A$ and has residue differing by $1$, so it is absent from this residue class. Consequently the root push/pop walk is exactly the reduced-signature stack after $A^{\mathrm{\prime }}$: push a removable $-$, pop on an addable $+$, implementing cancellation $-+$.

Earlier signature signs do not change whether $A^{\mathrm{\prime }}$ or any later push is removed. $A$ survives precisely when no later prefix has more pops than pushes; this is exactly root normality $\sigma (i,l)$. If it survives, the final stack consists of $A$ together with those above ordinary removable nodes in $A$'s residue class whose transposes survive as normal nodes. The stack proof of [F3] identifies this final stack with $N(i,l)$, so ${\nu }_a(i,l)=0$ if $A$ is not normal, and ${\nu }_a(i,l)=1+\mathrm{\#}\{\text{normal removable nodes strictly above }A\text{ of }A\text{’s residue}\}$ if $A$ is normal. The node count here is piecewise. It is not the unconditioned count of normal nodes weakly above $A$ when $A$ is nonnormal.

Work at $q=1$ with right Schur modules, reversing the star conversion used for the algebraic count. The Hecke presentation has $T_i^2=1$ and the symmetric-group braid relations; its standard basis $T_w$ maps bijectively to the group basis $w$, so $H(F,1,S_m)=F{S}_m$. Let $e$ be the weight projection of type $\omega =(1^m)$, with ${\mathrm{End}}_H(H)=H$ via ${\rho }_h(x)=hx$. The normalized semistandard map ${\phi }_{ST}$ is supported from $\mathrm{type}(T)$ to $\mathrm{type}(S)$. Thus $e{\phi }_{ST}e$ survives exactly when both types are $\omega$. These tableaux are ordinary standard tableaux, their defining fibers are singleton, and evaluation at $1$ gives ${\phi }_{st}\mapsto m_{st}$. The higher Schur ideal corner is exactly the higher Murphy ideal, as equality of the corresponding basis spans. The projection on a Weyl cell fixes $w_T$ exactly for $\mathrm{type}(T)=\omega$. To read its right $H$ action, choose the independent fixed outer index also of type $\omega$; evaluation at $1$ then turns ${\phi }_{st}{\rho }_h$ into $m_{st}h$ modulo the corresponding higher ideal. Hence $W^{\xi }e\cong C_{\mathrm{M}}(\xi )$ by $w_t\mapsto c_t$, with the actual same-shape Murphy action, over every field.

Choose all four indices $a,s,t,b$ standard of shape $\xi$ and type $\omega$ in the Schur cell product ${\phi }_{as}{\phi }_{tb}\equiv B_{\xi }(w_s,w_t){\phi }_{ab}$. Evaluate every map at $1$ in the corner. The Murphy product is $m_{as}{m}_{tb}\equiv B_{\xi }(w_s,w_t)m_{ab}$, so its independent cell-product coefficient gives $B_{\xi }(w_s,w_t)=G_{\mathrm{M},\xi }(c_s,c_t)$. The scalar is exactly $1$. Since $e^{\ast }=e$, contravariance gives $B_{\xi }(xe,y(1-e))=B_{\xi }(x,y(1-e)e)=0$. Thus the Weyl decomposition $W^{\xi }=W^{\xi }e\oplus W^{\xi }(1-e)$ is orthogonal, and ${\mathrm{rad}}_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}(W^{\xi }e)={\mathrm{rad}}_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}(W^{\xi })e$, including an identically zero corner form. The canonical shape/type $\xi$ diagonal is the projection onto $M^{\xi }$ and is idempotent, so the same cell-product coefficient gives canonical norm $1$; $L^{\xi }=W^{\xi }\mathrm{/}{\mathrm{rad}}_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}$ is nonzero simple. Projection by $e$ is exact (lift $ye$ by $xe$, and kernels are intersections fixed by $e$), so $L^{\xi }e\cong D_{\mathrm{M}}(\xi )$, including zero. At $q=1$ the unit form comparison [F2] gives $D_{\mathrm{M}}(\xi )\cong D^{{\xi }^{\mathrm{\prime }}}\otimes \mathrm{sgn}$ when ${\xi }^{\mathrm{\prime }}$ is regular, and zero otherwise by the published ordinary radical criterion. These nonzero quotients are distinctly labelled group simples. This proves restricted survival locally and uses no general finite-e classification.

Take $N=m=n$ and the tensorword realization of $S(N,m)$ already proved in [F3]. Let $p$ be the sum of all actual weight projections with last coordinate $1$. Its tensorwords have exactly one occurrence of $N$. Identify their remaining ordered positions with $(F^{N-1}{)}^{\otimes (m-1)}$. For each lower Schur orbit matrix $E_A$, define $\kappa (E_A)=E_{\mathrm{diag}(A,1)}$, where the last row and column have only their final entry $1$. On every word with one $N$ this map preserves its position and applies $E_A$ to the other positions. Consequently products are exactly the lower products, $\kappa (1)=p$, and $\kappa$ embeds $S(N-1,m-1)$ as a unital subalgebra of $pS(N,m)p$. This describes the lower Schur action on $Xp$, and for the star-converted left simple it is exactly the block-diagonal ${\mathrm{G}\mathrm{L}}_{N-1}$ action used in the character calculation, because $g^{\otimes (m-1)}$ expands in $E_A$ and $\mathrm{diag}(g,1)$ acts at the unchanged $N$ position. Transpose sends $E_A$ to $E_{A^{\mathsf{t}}}$, so $\kappa$ respects star. The first level counted above is therefore precisely $L_N(\mu )p$, converted to the right; no source-only definition of first level is being substituted.

Let $e^{\mathrm{\prime }}$ be the lower regular-type projection $(1^{N-1})$. In this embedding $\kappa (e^{\mathrm{\prime }})=e$, the full type $(1^N)$ projection. Use the explicit coset-word map $m_{\nu }{T}_d\mapsto (\text{row of entry }j\text{ in }{t}^{\nu }d{)}_j$. On the regular type its jth letter is $j{d}^{-1}$, in right permutation notation. Consequently ${\rho }_h:d\mapsto hd$ changes a regular word by the corresponding label permutation $h^{-1}$, because $(hd{)}^{-1}=d^{-1}{h}^{-1}$. A lower corner element ${\rho }_h^{\mathrm{\prime }}$ acts by this same label permutation $h^{-1}$ on $1,\dots ,N-1$ and fixes label $N$. The orbit embedding therefore gives exactly ${\rho }_h$ on the full regular tensorword basis, with no extra inversion or opposite-algebra map. Thus $\kappa ({\rho }_h^{\mathrm{\prime }})={\rho }_h$, with the subgroup action in the right convention exactly $X{\rho }_h$. Consequently $(Xp)e^{\mathrm{\prime }}=Xe$ as $F{S}_{N-1}$-modules, with the latter the ordinary restriction of $Xe$. Apply this to $X=L_N(\mu )$. This is the complete local $q=1$ first-level/corner identity; it is independent of the held generic comparison.

Apply the exact lower projection $e^{\mathrm{\prime }}$ to a finite composition series of $L_N(\mu )p$. Each factor $L_{N-1}(\eta )$ becomes $D_{\mathrm{M}}(\eta )$, proved by the lower-rank form/corner calculation of steps 14.1–15.1; it is a distinctly labelled simple when $\eta$ is restricted, and zero otherwise. Deleting repeated terms leaves a composition series of the group restriction corner. Thus the multiplicity of a restricted $\eta$ agrees exactly with its lower Schur multiplicity. Nonrestricted $\eta$ were retained in the earlier character calculation and are killed only here. For $\mu ={\lambda }^{\mathrm{\prime }}$, [F2] gives $L_N(\mu )e\cong D^{\lambda }\otimes \mathrm{sgn}$. The candidate ${\nu }_i=(\lambda \setminus A{)}^{\mathrm{\prime }}$ is restricted by the regular-removal hypothesis, and its image is $D^{\lambda \setminus A}\otimes \mathrm{sgn}$. Hence the desired group restriction multiplicity is $b_i$. Twisting both modules by sign preserves multiplicity because sign of $S_n$ restricts to sign of $S_{n-1}$. To match a left-module convention for the ordinary $D$ in the Statement, apply the exact inverse/star equivalence from right group modules to left ones, $g\cdot x=x{g}^{-1}$. It commutes with restriction and sign twisting and preserves composition series. The ordinary polytabloid quotient is the same partition-labelled quotient in this convention; if the dictionary uses its contragredient realization, the induced nondegenerate invariant quotient form identifies that simple with its contragredient. Explicitly $x\mapsto (y\mapsto \beta (x,y))$ is bijective and equivariant since $\beta (gx,y)=\beta (x,g^{-1}y)$. This uses the form only after passing to the nonzero radical quotient, as in Modular Specht form and radical quotient; no self-duality of a possibly degenerate Specht module is assumed. Together with the algebraic count and signature calculation, this gives both displayed cases of the theorem.

For $\lambda =(n)$ the ordinary simple is trivial and restricts trivially, so the unique removable node has multiplicity $1$. Its transposed $\mu =(1^n)$ node is normal: no subsequent same-residue addable can cancel it, and the preceding possible addable gives a $+-$ pair rather than a cancellation. There are no above ordinary removables. For $n=1$ this is the one-dimensional restriction to $S_0$; for $n=0$ the removable-node hypothesis is empty. These cases avoid requiring ${\mu }_N=0$ where it is false. All primes, all fields of characteristic $p$, all $p$-regular $\lambda$ and all regular removals retain the original stated scope.

The piecewise integer $\gamma (A)={\mathbf{1}}_{A\text{ normal}}$ times the number of normal nodes weakly above $A$ has value zero off normality and is at least one on normality, since $A$ itself is counted. The preceding multiplicity and boundary proofs therefore establish both the formula and its stated equivalence over every field of characteristic $p$.