Symmetric function

A symmetric function is a formal power series of bounded total degree that is invariant under every finite permutation of its variables. Symmetric functions form a graded ring with several useful bases.

Expand each factor as a geometric series indexed by a nonnegative matrix. The RSK correspondence preserves its row and column contents, changing the sum to pairs of Semistandard Young tableaux of the same shape and proving the displayed formal identity. Its kernel is also ${\sum }_{\lambda }{h}_{\lambda }(x)m_{\lambda }(y)={\sum }_{\lambda }{p}_{\lambda }(x)p_{\lambda }(y)/z_{\lambda }$. These are dual-basis kernels for the Hall inner product of symmetric functions, so the Schur expansion proves orthonormality. Applying the symmetric-function involution in one alphabet gives the dual identity ${\prod }_{i,j}(1+x_i{y}_j)={\sum }_{\lambda }{s}_{\lambda }(x)s_{{\lambda }^{\prime }}(y)$.

Given $F(t)={\sum }_{r\ge 0}{f}_r{t}^r$ with $f_0=1$, put $f_r=0$ for $r<0$. The Cauchy identity for symmetric functions and Jacobi–Trudi identity give ${\prod }_{i}F(t_i)={\sum }_{\lambda }{s}_{\lambda }^F{s}_{\lambda }(t)$, degree by degree in the completion. For $F(t)={\prod }_j(1-x_{j}t{)}^{-1}$ the coefficients are $s_{\lambda }(x)$; for $F(t)={\prod }_j(1+x_{j}t)$ they are $s_{{\lambda }^{\prime }}(x)$.

Apply the symmetric-function involution in one alphabet to the ordinary Cauchy identity. It changes complete generators to elementary ones and, by the dual Jacobi–Trudi identity, sends $s_{\lambda }$ to $s_{{\lambda }^{\prime }}$. This proves the identity as a formal series and relates pairings with transposed Young shapes to squarefree matrix weights.

The Fundamental theorem of symmetric polynomials gives freely generated elementary functions, so sending each $e_r$ to $h_r$ defines an algebra homomorphism. For $E(t)=\sum e_r{t}^r$ and $H(t)=\sum h_r{t}^r$, $E(-t)H(t)=1$. Applying the homomorphism shows $\omega (H(t))=E(t)$, hence ${\omega }^2=1$. Taking logarithms of $H$ gives the displayed action on power sums. It preserves the Hall inner product of symmetric functions because each power-sum basis vector acquires only a sign.

The coefficient of $m_{\mu }$ in $f_{\lambda }$ counts distinct rearrangements of the parts of $\lambda$ whose prefix-sum set contains the prefix sums of $\mu$. Indeed, Hall inner product of symmetric functions duality between $m_{\lambda },h_{\lambda }$ turns this coefficient into $(-1{)}^{|\lambda |-\ell (\lambda )}$ times the coefficient of $h_{\lambda }$ in $e_{\mu }$. Expanding $E(t)=H(-t{)}^{-1}$ expresses each $e_{{\mu }_j}$ as a signed sum of compositions of ${\mu }_j$. Concatenating these compositions gives precisely the counted rearrangements, and every sign is $(-1{)}^{|\lambda |-\ell (\lambda )}$.

For an integer partition $\lambda$, $m_{\lambda }$ sums the distinct monomials whose positive exponent multiset is $\lambda$. No multiplicity is attached to repeated parts. The functions of fixed total degree form a basis of the symmetric functions of that degree, since finite variable permutations identify precisely these monomial orbits.

Here $p_{\alpha }$ is a product of power-sum symmetric functions and $z_{\alpha }={\prod }_r{r}^{m_r}{m}_r!$ for part multiplicities $m_r$. The Frobenius characteristic map identifies this product with the ordinary character inner product. Schur functions are orthonormal, and the adjoint of multiplication by $p_k$ is $k\partial /\partial p_k$. These properties turn a power-sum multiplication formula into the Murnaghan–Nakayama rule.

The complete homogeneous symmetric polynomial $h_n$ is the sum of all monomials of total degree $n$. Its generating function is

The power-sum symmetric polynomial is $p_n={\sum }_i{x}_i^n$. For an integer partition $\mu$, write $p_{\mu }={\prod }_i{p}_{{\mu }_i}$.

The Schur polynomials form a basis of the ring of symmetric functions indexed by integer partitions.

Let $a_{\gamma }=\det [x_j^{{\gamma }_i}]$ and $\delta =(N-1,\dots ,0)$. Multiply the Jacobi–Trudi identity matrix by $B_{jk}=(-1{)}^{N-j}{e}_{N-j}(X\setminus x_k)$. The generating-function identity $H(t){\prod }_{i\ne k}(1-x_{i}t)=(1-x_{k}t{)}^{-1}$ makes the product matrix $[x_k^{{\lambda }_i+N-i}]$. Setting $\lambda =0$ shows $\det B=a_{\delta }$, proving the formula as a polynomial identity.

Put $b_i={\lambda }_i+N-i$. In the bialternant formula, substitute $x_j=e^{t{y}_j}$ with distinct $y_j$ and compare the first nonzero determinant coefficients as $t\to 0$. They arise from the distinct degrees $0,\dots ,N-1$ in the exponential expansions. The common factor in the $y_j$ and the factorials cancel, leaving the product of differences of the $b_i$ divided by the differences of $N-i$. This proves the displayed integral specialization without dividing by a vanishing Vandermonde value.

Sum the weight monomials of all Semistandard Young tableaux on the skew Young diagram $\lambda /\mu$, with weak rows and strict columns. A Bender-Knuth involution interchanges the counts of adjacent letters without changing the shape, so the sum is a symmetric function. Taking $\mu$ empty gives the Schur function.

The Jacobi–Trudi identity expresses a Schur polynomial asIn particular, $s_{(a,b)}=h_a{h}_b-h_{a+1}{h}_{b-1}$.

Here $[a]$ is the trivial character of $S_a$, $[0]$ is the induction-product unit, $[a]=0$ for $a<0$, and multiplication is induction of external tensor products. The Frobenius characteristic map turns the expression into the Jacobi–Trudi identity. A sign-reversing involution on intersecting lattice-path systems leaves precisely the nonintersecting systems corresponding to Semistandard Young tableaux, proving the determinant identity.

The Frobenius characteristic map sends class functions on $S_n$ to homogeneous symmetric functions of degree $n$. It satisfies

For a partition of an integer $\lambda ⊢n$, padded to $m$ rows, let $\delta =(m-1,\dots ,0)$. If a permutation has ${\alpha }_q$ cycles of length $q$, put $p_{\alpha }={\prod }_q({\sum }_i{x}_i^q{)}^{{\alpha }_q}$. The character of the Specht module $S^{\lambda }$ is the displayed coefficient. Equivalently $p_{\alpha }{A}_{\delta }={\sum }_{\lambda }{\chi }^{\lambda }(\alpha )A_{\lambda +\delta }$, the alternant form of the Frobenius characteristic map. It gives finite coefficient computations without constructing representation matrices.