The Frobenius Characteristic Map and the Symmetric-Function Dictionary

Intuition Beginner

Two subjects that look unrelated turn out to be the same subject in disguise. On one side sits the algebra of symmetric functions: polynomial-like expressions in many variables that stay the same when you shuffle the variables, met in the previous units. On the other side sits the study of how the shuffling group itself — the group of all ways to rearrange $n$ objects — can act on spaces. Each way of acting records a list of numbers, one number per kind of shuffle, called a character. The surprising fact is that there is a perfect dictionary between these character-lists and the symmetric functions.

Why would anyone want such a dictionary? Characters are hard to compute by hand and easy to get wrong. Symmetric functions, by contrast, come with a tidy algebra: you can multiply them, expand them in known building blocks, and read off answers. The dictionary lets you take a stubborn question about the shuffling group, translate it into a clean question about polynomials, solve it there, and translate the answer back. It is the same trick as using a map: the territory is hard to walk, but the map is easy to read.

The dictionary is called the Frobenius characteristic map. Its single most important entry is that the basic, irreducible character-lists — the genuinely indecomposable ways the shuffling group can act — correspond exactly to the Schur functions from the previous unit, one for each staircase shape. So the staircase shapes that index the building blocks of symmetric functions are the very same shapes that index the building blocks of group symmetry. This is the moment where all the tableau-counting of this chapter becomes representation theory.

Visual Beginner

The dictionary as a two-column correspondence. The left column lives in the world of the shuffling group on $n$ objects; the right column lives in the world of symmetric functions. The Frobenius map is the arrow translating left to right, and it never loses information.

world of shuffles (group side)	translates to (symmetric-function side)
the do-nothing action (everyone fixed)	the complete building block $h_n$
the sign action (track even vs. odd shuffles)	the elementary building block $e_n$
an irreducible action labelled by a shape	the Schur function $s$ of that shape
combining two group actions (induction)	multiplying the two symmetric functions
measuring overlap of two actions	measuring overlap of two functions

Reading the table top to bottom shows the dictionary respects every operation that matters. The do-nothing and sign actions go to the two simplest building blocks. The irreducible actions go to the Schur functions. Combining actions becomes multiplication, and the way you measure how much two actions overlap becomes the way you measure how much two symmetric functions overlap. A dictionary that preserves all this is the best kind: a question answered on one side is automatically answered on the other.

Worked example Beginner

Let us translate the two simplest actions of the shuffling group on three objects and watch the dictionary work. The group here rearranges $3$ objects, and its shuffles come in three kinds, sorted by how the rearrangement breaks into cycles: doing nothing (three fixed points), swapping two and fixing one (a single transposition), and cycling all three.

Step 1. The do-nothing action. This is the action that leaves everything put: every shuffle is recorded by the number $1$. Its character-list is the constant $1$ across all three kinds of shuffle.

Step 2. Apply the recipe. The Frobenius map averages the character-list against a power-sum bookkeeping symbol, one symbol per kind of shuffle, weighted by how common that kind is. Carrying out the weighted average for the constant list $1$ produces exactly the complete building block $h_3$. So the do-nothing action translates to $h_3$.

Step 3. The sign action. This records each shuffle by $+1$ if it is an even rearrangement and $-1$ if odd: doing nothing gives $+1$, a single swap gives $-1$, a three-cycle gives $+1$.

Step 4. Apply the recipe again. The same weighted average, now against the signed list, produces the elementary building block $e_3$. So the sign action translates to $e_3$.

Step 5. Read the dictionary. The two plainest actions landed on the two plainest building blocks, $h_3$ and $e_3$. The interesting actions — the irreducible ones — land on the Schur functions in between.

What this tells us: the recipe is a genuine translation that you can carry out by hand on small cases, and it sends recognizable group actions to recognizable symmetric functions. The plain actions give plain answers; the rich middle ground is where the Schur functions live, and that is the entry that makes the whole dictionary valuable.

Check your understanding Beginner

Formal definition Intermediate+

Work over the symmetric-function ring $\Lambda$ of 40.03.01 (notation $m_{\lambda },e_{\lambda },h_{\lambda },p_{\lambda },s_{\lambda },z_{\lambda },\omega ,⊵,{\lambda }^{\prime }$ inherited from there and from 40.03.02, all recorded in _meta/NOTATION.md). The representation-theoretic objects — the symmetric group $S_n$, its irreducible representations the Specht modules $S^{\lambda }$ with characters ${\chi }^{\lambda }$, induction $\mathrm{Ind}$, and the standard inner product on class functions — are imported from 07.05.01, 07.05.03 and applied here, not redeveloped.

Definition (the graded representation ring). For each $n\ge 0$ let $R^n=R(S_n)$ be the free $\mathbb{Z}$-module of virtual characters of $S_n$ (with $R^0=\mathbb{Z}$), i.e. $\mathbb{Z}$-linear combinations of irreducible characters; a class function on $S_n$ is an element of $R_{\mathbb{C}}^n=R^n\otimes \mathbb{C}$. Set $R={⨁}_{n\ge 0}{R}^n$. Equip $R$ with the induction product: for $f\in R^m$, $g\in R^n$, $$ f \cdot g ;=; \mathrm{Ind}{S_m \times S_n}^{S{m+n}}\big(f \otimes g\big) ;\in; R^{m+n}, $$ where $S_m\times S_n$ is the Young subgroup permuting $\{1,\dots ,m\}$ and $\{m+1,\dots ,m+n\}$ separately. This product is commutative, associative, and graded, making $R$ a graded commutative ring with unit $1\in R^0$. Equip each $R_{\mathbb{C}}^n$ with the character inner product $$ \langle f, g\rangle_{S_n} ;=; \frac{1}{n!}\sum_{\sigma \in S_n} f(\sigma),\overline{g(\sigma)} ;=; \sum_{\lambda \vdash n} z_\lambda^{-1}, f(\lambda),\overline{g(\lambda)}, $$ the second form grouping the average by conjugacy class: a permutation of cycle type $\lambda ⊢n$ has centralizer of order $z_{\lambda }$, so its class has $n!/z_{\lambda }$ elements. Under $⟨,{⟩}_{S_n}$ the irreducible characters $\{{\chi }^{\lambda }:\lambda ⊢n\}$ are orthonormal.

Definition (the Frobenius characteristic map). The Frobenius characteristic is the graded $\mathbb{Z}$-linear map $\mathrm{ch}:R\to \Lambda$ defined on $R^n$ by $$ \mathrm{ch}(f) ;=; \frac{1}{n!}\sum_{\sigma \in S_n} f(\sigma), p_{\rho(\sigma)} ;=; \sum_{\lambda \vdash n} z_\lambda^{-1}, f(\lambda), p_\lambda, $$ where $\rho (\sigma )$ is the cycle type of $\sigma$ and $f(\lambda )$ the common value of $f$ on the class of cycle type $\lambda$. Equivalently, $\mathrm{ch}$ is the linear map sending the class indicator $1_{\lambda }$ (the class function equal to $z_{\lambda }$ on the class $\lambda$ and $0$ elsewhere — chosen so that $⟨1_{\lambda },1_{\mu }⟩=z_{\lambda }{\delta }_{\lambda \mu }$) to the power sum $p_{\lambda }$. Because $\{z_{\lambda }^{-1}{p}_{\lambda }:\lambda ⊢n\}$ and the class functions $\{1_{\lambda }\}$ are dual bases on their respective sides, $\mathrm{ch}$ restricts to a $\mathbb{C}$-linear isomorphism $R_{\mathbb{C}}^n\stackrel{\sim }{\to }{\Lambda }_{\mathbb{C}}^n$ in each degree.

Counterexamples to common slips Intermediate+

• "$\mathrm{ch}$ is defined on representations, so it lands in $\mathbb{Z}$-combinations of Schur functions only." It is defined on all class functions once tensored with $\mathbb{C}$, and is a $\mathbb{C}$-isomorphism $R_{\mathbb{C}}^n\to {\Lambda }_{\mathbb{C}}^n$. The integral image $\mathrm{ch}(R^n)={⨁}_{\lambda }\mathbb{Z}{s}_{\lambda }$ recovers exactly the Schur lattice, but the power-sum formula makes sense for any class function, which is what lets it carry the Murnaghan-Nakayama inversion.

• "The product on $R$ is the tensor product of representations." It is induction, ${\mathrm{Ind}}_{S_m\times S_n}^{S_{m+n}}$, an external operation raising the rank from $m+n$ inputs; the pointwise tensor product of two characters of the same $S_n$ is a different (internal) product and corresponds under $\mathrm{ch}$ to the Hopf-algebraic Kronecker product, not to multiplication in $\Lambda$.

• "$\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$ is the definition of the Schur function." The Schur functions were defined combinatorially in 40.03.02 (the SSYT sum / bialternant / Jacobi-Trudi); that $\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$ is a theorem identifying two independently-constructed objects, and its content is that the Kostka matrix governs both sides.

Key theorem with proof Intermediate+

The signature theorem is that $\mathrm{ch}$ is not merely a linear bijection but an isometric ring isomorphism that sends irreducible characters to Schur functions — so the entire algebra and geometry of $\Lambda$ is the representation theory of the symmetric groups read in another alphabet.

Theorem (Frobenius characteristic is an isometric ring isomorphism, $\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$). The map $\mathrm{ch}:R\to \Lambda$ satisfies:

(a) (ring map) $\mathrm{ch}(f\cdot g)=\mathrm{ch}(f)\mathrm{ch}(g)$, carrying the induction product to multiplication in $\Lambda$;

(b) (isometry) $⟨f,g{⟩}_{S_n}=⟨\mathrm{ch}(f),\mathrm{ch}(g){⟩}_{\Lambda }$, the right side the Hall inner product of 40.03.03 under which $⟨p_{\lambda },p_{\mu }⟩=z_{\lambda }{\delta }_{\lambda \mu }$;

(c) (irreducibles to Schur) $\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$ for every $\lambda ⊢n$; in particular $\mathrm{ch}$ is a graded ring isomorphism $R_{\mathbb{C}}\stackrel{\sim }{\to }{\Lambda }_{\mathbb{C}}$ restricting to a $\mathbb{Z}$-isomorphism $R\stackrel{\sim }{\to }{⨁}_{\lambda }\mathbb{Z}{s}_{\lambda }=\Lambda$. ^{[Macdonald §7]}

Proof. (b) Isometry. Both inner products are diagonalised by cycle type. For $f,g\in R_{\mathbb{C}}^n$, $⟨f,g{⟩}_{S_n}={\sum }_{\lambda ⊢n}{z}_{\lambda }^{-1}f(\lambda )\overline{g(\lambda )}$ by grouping the average over $S_n$ into conjugacy classes, each of size $n!/z_{\lambda }$. On the symmetric-function side $\mathrm{ch}(f)={\sum }_{\lambda }{z}_{\lambda }^{-1}f(\lambda )p_{\lambda }$ and $\mathrm{ch}(g)={\sum }_{\mu }{z}_{\mu }^{-1}g(\mu )p_{\mu }$, so by bilinearity and $⟨p_{\lambda },p_{\mu }{⟩}_{\Lambda }=z_{\lambda }{\delta }_{\lambda \mu }$, $$ \langle \mathrm{ch}(f), \mathrm{ch}(g)\rangle_\Lambda = \sum_{\lambda,\mu} z_\lambda^{-1} z_\mu^{-1} f(\lambda)\overline{g(\mu)}, \langle p_\lambda, p_\mu\rangle = \sum_\lambda z_\lambda^{-1} f(\lambda)\overline{g(\lambda)} = \langle f, g\rangle_{S_n}. $$

(a) Ring map. It suffices to check on a spanning set; take $f=1_{\mu }^{(m)}$ and $g=1_{\nu }^{(n)}$, the class indicators with $\mathrm{ch}(1_{\mu })=p_{\mu }$. By transitivity of induction and the description of conjugacy classes of $S_m\times S_n$ inside $S_{m+n}$, the induced class function ${\mathrm{Ind}}_{S_m\times S_n}^{S_{m+n}}(1_{\mu }\otimes 1_{\nu })$ is supported on permutations whose cycle type is the union (concatenation) $\mu \cup \nu$, with the value computed by Frobenius reciprocity. Tracking the constants, $\mathrm{ch}(1_{\mu }\cdot 1_{\nu })=p_{\mu \cup \nu }=p_{\mu }{p}_{\nu }=\mathrm{ch}(1_{\mu })\mathrm{ch}(1_{\nu })$, since $p_{\mu \cup \nu }={\prod }_i{p}_{(\mu \cup \nu {)}_i}$ multiplies the two power-sum products. Because the $1_{\mu }$ span $R_{\mathbb{C}}$ and $\mathrm{ch}$ is linear, multiplicativity holds throughout.

(c) $\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$. Consider the permutation character ${\eta }^{\mu }={\mathrm{Ind}}_{S_{\mu }}^{S_n}1$ of the Young subgroup $S_{\mu }=S_{{\mu }_1}\times \cdots \times S_{{\mu }_{\ell }}$. By (a) and $\mathrm{ch}(1_{S_k})=h_k$ (the all-ones character of $S_k$ has $\mathrm{ch}(1)={\sum }_{\rho ⊢k}{z}_{\rho }^{-1}{p}_{\rho }=h_k$, by the power-sum expansion of 40.03.01), one gets $\mathrm{ch}({\eta }^{\mu })=h_{\mu }$. Young's rule decomposes ${\eta }^{\mu }={\sum }_{\lambda }{K}_{\lambda \mu }{\chi }^{\lambda }$ with $K_{\lambda \mu }$ the Kostka numbers, so applying $\mathrm{ch}$ gives $h_{\mu }={\sum }_{\lambda }{K}_{\lambda \mu }\mathrm{ch}({\chi }^{\lambda })$. Meanwhile 40.03.02 gives $h_{\mu }={\sum }_{\lambda }{K}_{\lambda \mu }{s}_{\lambda }$ (the dual of the Kostka/SSYT expansion $s_{\lambda }={\sum }_{\mu }{K}_{\lambda \mu }{m}_{\mu }$ under the Hall pairing $⟨h_{\mu },s_{\lambda }⟩=K_{\lambda \mu }$). The Kostka matrix $(K_{\lambda \mu })$ is unitriangular against dominance, hence invertible over $\mathbb{Z}$, so the two expansions force $\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$ term by term. By (b) the orthonormal basis $\{{\chi }^{\lambda }\}$ maps to an orthonormal set, and $\{s_{\lambda }\}$ is a $\mathbb{Z}$-basis of $\Lambda$, so $\mathrm{ch}$ is an isometric ring isomorphism. $\square$

Bridge. This theorem is the foundational reason the combinatorics of the previous units is representation theory: the SSYT sum, the bialternant, and the Jacobi-Trudi determinant that defined $s_{\lambda }$ in 40.03.02 are now three formulas for the irreducible character ${\chi }^{\lambda }$, and the Hall inner product of 40.03.03 is exactly the $S_n$-character inner product. It builds toward the Murnaghan-Nakayama rule and the Littlewood-Richardson rule of Advanced results, where character values and tensor decompositions become power-sum and product expansions in $\Lambda$. It appears again in 07.05.10, where the same identity $p_{\mu }={\sum }_{\lambda }{\chi }^{\lambda }(\mu )s_{\lambda }$ is read as a recursive character computation. The central insight is that induction on the group side is multiplication on the symmetric-function side; this is exactly why the orthonormality of irreducible characters and the orthonormality of Schur functions are one statement seen through the isometry, and putting these together, $\mathrm{ch}$ generalises every Schur identity into a character identity and conversely. The bridge is that $\Lambda$ and ${⨁}_{n}R(S_n)$ are the same graded ring.

Exercises Intermediate+

Advanced results Master

The characteristic map converts the four structural pillars of symmetric-function theory into the four structural pillars of symmetric-group representation theory, and the dictionary is exact in each.

Theorem 1 (Schur orthonormality is character orthonormality). Under the isometry (b) of the Key theorem, the Hall orthonormality $⟨s_{\lambda },s_{\mu }{⟩}_{\Lambda }={\delta }_{\lambda \mu }$ of 40.03.02, 40.03.03 is the orthonormality $⟨{\chi }^{\lambda },{\chi }^{\mu }{⟩}_{S_n}={\delta }_{\lambda \mu }$ of the irreducible characters 07.05.03. The Cauchy identity ${\prod }_{i,j}(1-x_i{y}_j{)}^{-1}={\sum }_{\lambda }{s}_{\lambda }(x)s_{\lambda }(y)$, proved bijectively by RSK in 40.03.04, is the symmetric-function image of the second orthogonality relation ${\sum }_{\lambda }{\chi }^{\lambda }(\sigma ){\chi }^{\lambda }(\tau )=z_{\rho (\sigma )}{\delta }_{\rho (\sigma ),\rho (\tau )}$, since pairing the Cauchy kernel against ${\mathrm{ch}}^{-1}$ in each variable returns the class-function reproducing kernel ${\sum }_{\lambda }{1}_{\lambda }\otimes z_{\lambda }^{-1}{1}_{\lambda }$ ^{[Macdonald §7]}.

Theorem 2 (Littlewood-Richardson is induction of irreducibles). The structure constants of $\Lambda$ in the Schur basis, $s_{\mu }{s}_{\nu }={\sum }_{\lambda }{c}_{\mu \nu }^{\lambda }{s}_{\lambda }$ with $c_{\mu \nu }^{\lambda }$ the Littlewood-Richardson coefficients of 40.03.05, are the multiplicities in the induced representation: $$ \mathrm{Ind}{S_m \times S_n}^{S{m+n}}\big(S^\mu \boxtimes S^\nu\big) ;\cong; \bigoplus_{\lambda \vdash m+n} \big(S^\lambda\big)^{\oplus c^\lambda_{\mu\nu}}. $$ This is immediate from (a): $\mathrm{ch}({\chi }^{\mu }\cdot {\chi }^{\nu })=s_{\mu }{s}_{\nu }={\sum }_{\lambda }{c}_{\mu \nu }^{\lambda }{s}_{\lambda }=\mathrm{ch}\left({\sum }_{\lambda }{c}_{\mu \nu }^{\lambda }{\chi }^{\lambda }\right)$, and $\mathrm{ch}$ is injective. The non-negativity of the $c_{\mu \nu }^{\lambda }$ — a fact requiring real combinatorial work on the symmetric-function side (the LR rule counts LR tableaux) — is automatic on the representation side, where multiplicities of irreducibles in an actual representation are non-negative integers by construction.

Theorem 3 (Murnaghan-Nakayama and the Frobenius formula). Inverting $\mathrm{ch}$ on power sums expresses $p_{\mu }$ in the Schur basis with character values as coefficients: $$ p_\mu ;=; \sum_{\lambda \vdash n} \chi^\lambda(\mu), s_\lambda, \qquad \text{equivalently} \qquad \chi^\lambda(\mu) = \langle p_\mu, s_\lambda\rangle_\Lambda. $$ Expanding $p_{\mu }=p_{{\mu }_1}{p}_{{\mu }_2}\cdots$ one $p_{{\mu }_i}$ at a time and applying the power-sum Pieri rule $p_r{s}_{\nu }={\sum }_{\xi }(-1{)}^{\mathrm{ht}(\xi /\nu )}{s}_{\xi }$ over border strips $\xi /\nu$ of size $r$ yields the Murnaghan-Nakayama rule 07.05.10: ${\chi }^{\lambda }(\mu )={\sum }_T(-1{)}^{\mathrm{ht}(T)}$ over border-strip tableaux $T$ of shape $\lambda$ and type $\mu$. Specialising to the longest hook expansions recovers the original Frobenius character formula ${\chi }^{\lambda }(\mu )=[x^{\lambda +\delta }]a_{\delta }{p}_{\mu }$, the coefficient extraction in the bialternant numerator ^{[Frobenius 1900]}. The dimension $f^{\lambda }={\chi }^{\lambda }(1^n)=⟨p_1^n,s_{\lambda }⟩$ is the hook-length formula $n!/{\prod }_{(i,j)}h(i,j)$ of 40.03.02.

Theorem 4 (Schur-Weyl duality through $\mathrm{ch}$). The bialternant side $s_{\lambda }=a_{\lambda +\delta }/a_{\delta }$ of 40.03.02 is the irreducible ${\mathrm{GL}}_N$-character of highest weight $\lambda$, while the Frobenius side $\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$ makes $s_{\lambda }$ the irreducible $S_n$-character; the single function $s_{\lambda }$ being both is the symmetric-function shadow of Schur-Weyl duality 07.05.04, the commuting actions of ${\mathrm{GL}}_N$ and $S_n$ on $({\mathbb{C}}^N{)}^{\otimes n}$ giving the decomposition $({\mathbb{C}}^N{)}^{\otimes n}={⨁}_{\lambda }{V}_{\lambda }^{\mathrm{GL}}\otimes S^{\lambda }$. Taking the bigraded character and applying $\mathrm{ch}$ in the $S_n$ variable turns the duality into the Cauchy identity, so the two centralizing algebras are the two factorisations of one symmetric kernel.

Synthesis. Putting these together, the foundational reason symmetric functions and symmetric-group representations are interchangeable is that $\mathrm{ch}$ is an isometric isomorphism of graded rings, and every structure on either side is forced to match. The central insight is that induction on ${⨁}_{n}R(S_n)$ is exactly multiplication in $\Lambda$, so the Littlewood-Richardson coefficients of 40.03.05 are induced-representation multiplicities and their non-negativity — hard combinatorics on the $\Lambda$ side — is automatic on the group side; this is exactly why the orthonormality of Schur functions, proved analytically through the Cauchy identity and RSK in 40.03.04, coincides with the orthonormality of irreducible characters. The Murnaghan-Nakayama rule is dual to the Pieri rules under $\mathrm{ch}$: the complete-function Pieri rule of 40.03.02 is the branching of permutation modules, while the power-sum Pieri rule is the border-strip character recursion of 07.05.10, and the two are exchanged by passing between the $h$-basis and the $p$-basis of 40.03.01. The bialternant being simultaneously a ${\mathrm{GL}}_N$- and an $S_n$-character generalises into Schur-Weyl duality 07.05.04, so the whole determinantal calculus of 40.03.02 reappears as the character theory of 07.05.03. The bridge is that $\mathrm{ch}$ identifies the Hall inner product of 40.03.03 with the character inner product, making the Schur basis the orthonormal eigenbasis on both sides at once, and the RSK correspondence of 40.03.04 the bijective proof that the identification is exact.

Full proof set Master

Proposition 1 (the isometry, in full). For $f,g\in R_{\mathbb{C}}^n$, $⟨f,g{⟩}_{S_n}=⟨\mathrm{ch}(f),\mathrm{ch}(g){⟩}_{\Lambda }$.

Proof. The conjugacy classes of $S_n$ are indexed by cycle types $\lambda ⊢n$; the class of type $\lambda$ has cardinality $n!/z_{\lambda }$, where $z_{\lambda }={\prod }_i{i}^{m_i}{m}_i!$ is the centralizer order (40.03.01). A class function $f$ is constant on each class, with value $f(\lambda )$, so $$ \langle f, g\rangle_{S_n} = \frac{1}{n!}\sum_{\sigma \in S_n} f(\sigma)\overline{g(\sigma)} = \frac{1}{n!}\sum_{\lambda \vdash n} \frac{n!}{z_\lambda} f(\lambda)\overline{g(\lambda)} = \sum_{\lambda \vdash n} z_\lambda^{-1} f(\lambda)\overline{g(\lambda)}. $$ On the symmetric-function side, $\mathrm{ch}(f)={\sum }_{\lambda }{z}_{\lambda }^{-1}f(\lambda )p_{\lambda }$ and the Hall inner product of 40.03.03 satisfies $⟨p_{\lambda },p_{\mu }{⟩}_{\Lambda }=z_{\lambda }{\delta }_{\lambda \mu }$. Bilinearity gives $$ \langle \mathrm{ch}(f), \mathrm{ch}(g)\rangle_\Lambda = \sum_{\lambda, \mu} z_\lambda^{-1} z_\mu^{-1} f(\lambda)\overline{g(\mu)},\langle p_\lambda, p_\mu\rangle_\Lambda = \sum_\lambda z_\lambda^{-2} f(\lambda)\overline{g(\lambda)}, z_\lambda = \sum_\lambda z_\lambda^{-1} f(\lambda)\overline{g(\lambda)}, $$ which is the displayed value of $⟨f,g{⟩}_{S_n}$. The two agree. $\square$

Proposition 2 (multiplicativity). $\mathrm{ch}(f\cdot g)=\mathrm{ch}(f)\mathrm{ch}(g)$ for the induction product on $R$.

Proof. By linearity it suffices to verify on the class-indicator basis $1_{\mu }\in R_{\mathbb{C}}^m$ (value $z_{\mu }$ on the class of type $\mu$, else $0$), for which $\mathrm{ch}(1_{\mu })=z_{\mu }^{-1}{z}_{\mu }{p}_{\mu }=p_{\mu }$. Compute the induction product $1_{\mu }\cdot 1_{\nu }={\mathrm{Ind}}_{S_m\times S_n}^{S_{m+n}}(1_{\mu }\otimes 1_{\nu })$ via Frobenius reciprocity against an arbitrary class indicator $1_{\xi }$ of $S_{m+n}$: $$ \langle \mathbf{1}\mu \cdot \mathbf{1}\nu, \mathbf{1}\xi\rangle{S_{m+n}} = \langle \mathbf{1}\mu \otimes \mathbf{1}\nu, \mathrm{Res},\mathbf{1}\xi\rangle{S_m \times S_n}. $$ The restriction of $1_{\xi }$ to $S_m\times S_n$ is supported on pairs $(\sigma ,\tau )$ whose cycle types ${\mu }^{\prime },{\nu }^{\prime }$ concatenate to $\xi$, i.e. ${\mu }^{\prime }\cup {\nu }^{\prime }=\xi$. Tracking the centralizer constants, the pairing is nonzero precisely when $\mu \cup \nu =\xi$, and equals the value forcing $1_{\mu }\cdot 1_{\nu }=(\text{const})1_{\mu \cup \nu }$ with $\mathrm{ch}(1_{\mu }\cdot 1_{\nu })=p_{\mu \cup \nu }$. Since $p_{\mu \cup \nu }=p_{\mu }{p}_{\nu }$ (concatenation of cycle types multiplies the power-sum products), $\mathrm{ch}(1_{\mu }\cdot 1_{\nu })=p_{\mu }{p}_{\nu }=\mathrm{ch}(1_{\mu })\mathrm{ch}(1_{\nu })$. The $1_{\mu }$ span $R_{\mathbb{C}}$, so multiplicativity holds on all of $R_{\mathbb{C}}$. $\square$

Proposition 3 ($\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$). For every $\lambda ⊢n$, the characteristic of the irreducible Specht character is the Schur function.

Proof. For a composition (or partition) $\mu \models n$ let $S_{\mu }=S_{{\mu }_1}\times \cdots \times S_{{\mu }_{\ell }}$ be the Young subgroup and ${\eta }^{\mu }={\mathrm{Ind}}_{S_{\mu }}^{S_n}{1}_{S_{\mu }}$ the permutation character on cosets. The all-ones character of $S_k$ has $\mathrm{ch}(1_{S_k})={\sum }_{\rho ⊢k}{z}_{\rho }^{-1}{p}_{\rho }=h_k$ by the power-sum expansion of $h_k$ from 40.03.01, so by Proposition 2 and the factorisation ${\eta }^{\mu }=1_{S_{{\mu }_1}}\cdots 1_{S_{{\mu }_{\ell }}}$ under induction product, $\mathrm{ch}({\eta }^{\mu })=h_{{\mu }_1}\cdots h_{{\mu }_{\ell }}=h_{\mu }$. Young's rule (the decomposition of permutation modules of Young subgroups, 07.05.03) gives ${\eta }^{\mu }={\sum }_{\lambda }{K}_{\lambda \mu }{\chi }^{\lambda }$ with $K_{\lambda \mu }$ the Kostka numbers and $K_{\lambda \lambda }=1$, $K_{\lambda \mu }=0$ unless $\lambda ⊵\mu$. Apply $\mathrm{ch}$: $h_{\mu }={\sum }_{\lambda }{K}_{\lambda \mu }\mathrm{ch}({\chi }^{\lambda })$. On the symmetric-function side, 40.03.02 gives the dual Kostka relation $h_{\mu }={\sum }_{\lambda }{K}_{\lambda \mu }{s}_{\lambda }$ (from $⟨h_{\mu },s_{\lambda }⟩=⟨h_{\mu },{\sum }_{\nu }{K}_{\lambda \nu }{m}_{\nu }⟩=K_{\lambda \mu }$ and self-duality of $\{s_{\lambda }\}$). Subtracting, ${\sum }_{\lambda }{K}_{\lambda \mu }(\mathrm{ch}({\chi }^{\lambda })-s_{\lambda })=0$ for all $\mu ⊢n$. The matrix $(K_{\lambda \mu })$ is unitriangular against dominance, hence invertible over $\mathbb{Z}$, so the bracketed differences vanish: $\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$. $\square$

Proposition 4 (inversion on power sums; Murnaghan-Nakayama identity). $p_{\mu }={\sum }_{\lambda ⊢n}{\chi }^{\lambda }(\mu )s_{\lambda }$, and ${\chi }^{\lambda }(\mu )=⟨p_{\mu },s_{\lambda }{⟩}_{\Lambda }$.

Proof. The power sum $p_{\mu }=\mathrm{ch}(1_{\mu })$ is the characteristic of the class indicator $1_{\mu }$ (Proposition 2's basis). Expand $1_{\mu }$ in the orthonormal basis of irreducible characters: $1_{\mu }={\sum }_{\lambda }⟨1_{\mu },{\chi }^{\lambda }{⟩}_{S_n}{\chi }^{\lambda }$. The coefficient is $⟨1_{\mu },{\chi }^{\lambda }{⟩}_{S_n}={\sum }_{\rho }{z}_{\rho }^{-1}{1}_{\mu }(\rho )\overline{{\chi }^{\lambda }(\rho )}=z_{\mu }^{-1}\cdot z_{\mu }\cdot {\chi }^{\lambda }(\mu )={\chi }^{\lambda }(\mu )$ (real character values). Apply $\mathrm{ch}$, using $\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$ from Proposition 3: $p_{\mu }=\mathrm{ch}(1_{\mu })={\sum }_{\lambda }{\chi }^{\lambda }(\mu )\mathrm{ch}({\chi }^{\lambda })={\sum }_{\lambda }{\chi }^{\lambda }(\mu )s_{\lambda }$. Pairing with $s_{\lambda }$ and using $⟨s_{\lambda },s_{\nu }⟩={\delta }_{\lambda \nu }$ inverts the relation: $⟨p_{\mu },s_{\lambda }{⟩}_{\Lambda }={\chi }^{\lambda }(\mu )$. Iterating the power-sum Pieri rule $p_r{s}_{\nu }={\sum }_{\xi }(-1{)}^{\mathrm{ht}(\xi /\nu )}{s}_{\xi }$ (sum over border strips of size $r$, proved via the bialternant in 40.03.02) on $p_{\mu }=p_{{\mu }_1}\cdots p_{{\mu }_{\ell }}$ and reading the coefficient of $s_{\lambda }$ produces the border-strip-tableau sum, the Murnaghan-Nakayama rule of 07.05.10. $\square$

Proposition 5 (Littlewood-Richardson as induction multiplicity). ${\mathrm{Ind}}_{S_m\times S_n}^{S_{m+n}}(S^{\mu }⊠S^{\nu })\cong {⨁}_{\lambda }(S^{\lambda }{)}^{\oplus c_{\mu \nu }^{\lambda }}$ with $c_{\mu \nu }^{\lambda }$ the Littlewood-Richardson coefficients.

Proof. The induced module has character ${\chi }^{\mu }\cdot {\chi }^{\nu }$ (induction product). By Proposition 2 and Proposition 3, $\mathrm{ch}({\chi }^{\mu }\cdot {\chi }^{\nu })=\mathrm{ch}({\chi }^{\mu })\mathrm{ch}({\chi }^{\nu })=s_{\mu }{s}_{\nu }$. The Schur expansion $s_{\mu }{s}_{\nu }={\sum }_{\lambda }{c}_{\mu \nu }^{\lambda }{s}_{\lambda }$ defines the Littlewood-Richardson coefficients of 40.03.05. Since $\mathrm{ch}$ is a $\mathbb{Z}$-isomorphism onto the Schur lattice (Key theorem (c)), ${\mathrm{ch}}^{-1}(s_{\lambda })={\chi }^{\lambda }$ gives ${\chi }^{\mu }\cdot {\chi }^{\nu }={\sum }_{\lambda }{c}_{\mu \nu }^{\lambda }{\chi }^{\lambda }$. The multiplicity of the irreducible ${\chi }^{\lambda }$, namely $⟨{\chi }^{\mu }\cdot {\chi }^{\nu },{\chi }^{\lambda }{⟩}_{S_{m+n}}=c_{\mu \nu }^{\lambda }$, is the multiplicity of $S^{\lambda }$ in the induced module, giving the stated decomposition. The coefficients are non-negative because they count multiplicities of irreducibles in an actual representation. $\square$

Connections Master

• This unit is the capstone of the symmetric-functions chapter, resting directly on the Schur functions of 40.03.02: the combinatorial, bialternant, and Jacobi-Trudi descriptions of $s_{\lambda }$ proved there become three formulas for the irreducible $S_n$-character ${\chi }^{\lambda }$, the Pieri rules become the branching of permutation and induced modules, and the Kostka unitriangularity is the engine of the proof $\mathrm{ch}({\chi }^{\lambda })=s_{\lambda }$. The complete and elementary functions and the power-sum expansion $h_n={\sum }_{\rho }{z}_{\rho }^{-1}{p}_{\rho }$ of 40.03.01 give $\mathrm{ch}$ of the all-ones and sign characters, and the involution $\omega$ of 40.03.01, being $s_{\lambda }\mapsto s_{{\lambda }^{\prime }}$, corresponds under $\mathrm{ch}$ to tensoring a representation with the sign character.

• The isometry half of the theorem is exactly the Hall inner product of 40.03.03 read as the $S_n$-character inner product: $⟨s_{\lambda },s_{\mu }⟩={\delta }_{\lambda \mu }$ is $⟨{\chi }^{\lambda },{\chi }^{\mu }⟩={\delta }_{\lambda \mu }$, and the Cauchy identity equivalent to it is the second character orthogonality relation. The RSK correspondence of 40.03.04 supplies the bijective proof of the Cauchy identity, so RSK is the combinatorial certificate that $\mathrm{ch}$ is an isometry; the Littlewood-Richardson coefficients of 40.03.05 are, through this unit, the multiplicities of irreducibles in ${\mathrm{Ind}}_{S_{\mu }\times S_{\nu }}^{S_{m+n}}$, and their manifest non-negativity on the representation side is what the LR combinatorial rule must reproduce.

• The map links the chapter to the symmetric-group representation theory of 07.05.01–07.05.04 and 07.05.10: the Specht modules $S^{\lambda }$ and their characters of 07.05.03 are the objects $\mathrm{ch}$ carries to Schur functions, the Young-diagram and hook-length combinatorics of 07.05.02 computes their dimensions $f^{\lambda }$, the Murnaghan-Nakayama rule of 07.05.10 is the power-sum inversion $p_{\mu }={\sum }_{\lambda }{\chi }^{\lambda }(\mu )s_{\lambda }$, and Schur-Weyl duality of 07.05.04 is the statement that the single function $s_{\lambda }$ is simultaneously a ${\mathrm{GL}}_N$- and an $S_n$-character. The graded-ring self-duality that makes $\mathrm{ch}$ an isomorphism is the Hopf-algebra structure on $\Lambda$ from 40.03.01, placing ${⨁}_{n}R(S_n)$ and $\Lambda$ as the same positive self-dual Hopf algebra.

Historical & philosophical context Master

The identification of symmetric-group characters with symmetric functions originates with Georg Frobenius, who in his 1900 paper Über die Charaktere der symmetrischen Gruppe (Sitzungsberichte der Preussischen Akademie) computed the irreducible characters ${\chi }^{\lambda }(\mu )$ as the coefficient of the monomial $x^{\lambda +\delta }$ in the product of the Vandermonde $a_{\delta }$ with the power sum $p_{\mu }$ ^{[Frobenius 1900]}; this determinantal formula is the inverse of the characteristic map read on power sums, $p_{\mu }={\sum }_{\lambda }{\chi }^{\lambda }(\mu )s_{\lambda }$. Issai Schur's 1901 Berlin dissertation ^{[Schur 1901]} identified the bialternant $a_{\lambda +\delta }/a_{\delta }$ with the polynomial irreducible character of ${\mathrm{GL}}_n(\mathbb{C})$, so that the same Schur function names a ${\mathrm{GL}}_n$- and (through Frobenius) an $S_n$-irreducible — the algebraic seed of what Schur later developed as the duality between the two groups.

The recursive evaluation of ${\chi }^{\lambda }(\mu )$ through border strips was given by Francis Murnaghan (1937) and Tadasi Nakayama (1940), and the modern packaging of the whole correspondence as an isometric isomorphism of the graded ring $R={⨁}_{n}R(S_n)$ onto $\Lambda$ is due to the school that reorganised the subject around the inverse-limit ring: Philip Hall and, in the explicit characteristic-map form, Macdonald's Symmetric Functions and Hall Polynomials (1979; 2nd ed. 1995). The recognition that the induction-product ring of all symmetric groups is a self-dual Hopf algebra with the irreducibles as a distinguished positive orthonormal basis was made precise by Geissinger (1977) and Zelevinsky (1981) ^{[Zelevinsky 1981]}, which situates the Frobenius map as the canonical isomorphism between the two universal positive self-dual Hopf algebras attached to the symmetric groups and to $\Lambda$.

Bibliography Master

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