The symmetric functions catalog

An overview of symmetric functions and related topics

2020-10-02

Symmetric functions

A function $f$ in $n$ variables is symmetric if for any permutation $w\in \symS_n,$ we have

\[ f(x_1,x_2,\dotsc,x_n) = f(x_{w(1)},x_{w(2)},\dotsc,x_{w(n)}). \]

Below we list the most common families symmetric functions. These are all commonly used bases for the space of symmetric functions.

Here are a few great resources:

Monomial symmetric functions

Given a partition $\lambda,$ we define the monomial symmetric functions as

\[ \monomial_\lambda(\xvec) = \sum_{\alpha \sim \lambda} \xvec^{\alpha} \]

where $\alpha \sim \lambda$ if the parts of $\alpha$ is a rearrangement of the parts of $\lambda.$

The augmented monomial symmetric functions are defined as $\tilde{\monomial}_\lambda \coloneqq m_1!m_2!\dotsm m_n! \monomial_\lambda$ where $\lambda = (1^{m_1},2^{m_2},\dotsc).$ See [Mer15] for more background.

Elementary symmetric functions

The elementary symmetric functions are defined as follows:

\[ \elementaryE_k(\xvec) = \sum_{i_1 \lt i_2 \lt \dotsb \lt i_k } x_{i_1} \dotsm x_{i_k} = \monomial_{1^k}(\xvec), \qquad \elementaryE_\lambda(\xvec) = \elementaryE_{\lambda_1} \elementaryE_{\lambda_2} \dotsm \elementaryE_{\lambda_\ell} \]

Complete homogeneous symmetric functions

Similar to the elementary symmetric functions, the complete homogeneous functions are defined as

\[ \completeH_k(\xvec) = \sum_{i_1 \leq i_2 \leq \dotsb \leq i_k } x_{i_1} \dotsm x_{i_k} = \sum_{\lambda \vdash k} \monomial_\lambda(\xvec), \qquad \completeH_\lambda(\xvec) = \completeH_{\lambda_1} \completeH_{\lambda_2} \dotsm \completeH_{\lambda_\ell} \]

Powersum symmetric functions

The powersum symmetric functions are defined as

\[ \powerSum_k(\xvec) = x_{1}^k + x_2^k + x_3^k + \dotsb = \monomial_{(k)}(\xvec), \qquad \powerSum_\lambda(\xvec) = \powerSum_{\lambda_1} \powerSum_{\lambda_2} \dotsm \powerSum_{\lambda_\ell}. \]

They serve as an orthogonal basis for the Hall inner product. We have the product expansions [Sec. 1.4, Mac95],

\[ \prod_{i,j\geq 1}(1-x_i y_j)^{-1} = \sum_{\lambda} z_\lambda^{-1} \powerSum_\lambda(\xvec)\powerSum_\lambda(\yvec). \]

and

\[ \prod_{i,j\geq 1}(1+x_i y_j) = \sum_{\lambda} (-1)^{|\lambda|-\length(\lambda)} z_\lambda^{-1} \powerSum_\lambda(\xvec)\powerSum_\lambda(\yvec). \]
Lemma. \[ \prod_{i,j,k\geq 1}(1+x_i y_j z_k) = \sum_{\lambda} (-1)^{|\lambda|-\length(\lambda)} z_\lambda^{-1} \powerSum_\lambda(\xvec)\powerSum_\lambda(\yvec)\powerSum_\lambda(\zvec). \]
Proof.

First note that

\begin{align} \prod_{i,j,k\geq 1}(1+x_i y_j z_k) &= \prod_{k\geq 1} \prod_{i,j\geq 1}(1+x_i (y_j z_k)) \\ &= \prod_{k\geq 1} \sum_{\lambda} (-1)^{|\lambda|-\length(\lambda)} z_\lambda^{-1} \powerSum_\lambda(\xvec)\powerSum_\lambda(\yvec z_k). \end{align}

Now using that $\powerSum_\lambda(\yvec z_k) = z_k^{|\lambda|} \powerSum_\lambda(\yvec),$ we can easily deduce the statement.

There are two quasisymmetric refinements of the powersum symmetric functions given in [BDHMN17].

Conjecture (Sundaram 2018, [Sun18]).

Let $L_n$ be the reverse lexicographic ordering of partitions. Then the following sum over any initial segment,

\[ \sum_{\lambda \in [1^n,\mu]} \powerSum_\lambda(x) \]

is Schur-positive.

For some partial progress on S. Sundaram's conjecture, see these FPSAC 2019 proceedings.

Forgotten symmetric functions

Finally, there is a family of symmetric functions, $\forgotten_\lambda(\xvec),$ defined as $\forgotten_\lambda = \omega(\monomial_\lambda)$ (sometimes $(-1)^{|\lambda|-\length(\lambda)}\omega(\monomial_\lambda)$) where $\omega$ is the involution on symmetric functions. Note that $\langle \forgotten_\lambda, \elementaryE_\mu \rangle = \delta_{\lambda \mu}.$

It does not have as many applications as the other bases.

In [Ex. 7.9, Sta01], the following monomial expansion is given, where $\lambda \vdash n$ has $\ell$ parts:

\begin{equation}\label{eq:forgottenInMonomial} (-1)^{n-\ell} \forgotten_\lambda(\xvec) = \sum_{\mu} a_{\lambda\mu} \monomial_\mu(\xvec) \end{equation}

where $a_{\lambda \mu}$ is the number of distinct permutations $(\alpha_1,\dotsc,\alpha_\ell)$ of $(\lambda_1,\dotsc,\lambda_\ell)$ such that

\[ \{ \alpha_1+ \dotsb + \alpha_i : i \in [\ell] \} \supseteq \{ \mu_1+ \dotsb + \mu_j : j \in \length(\mu) \}. \]

Various identities

We have that

\[ \sum_{i=0}^n (-1)^i \elementaryE_i \completeH_{n-i} = \delta_{n,0}. \]

Introducing

\[ E(z) = \sum_{k \geq 0} \elementaryE_k z^k \text{ and } H(z) = \sum_{k \geq 0} \completeH_k z^k \]

we get $E(-z)H(z)=1.$ By letting $\gamma_k \coloneqq \powerSum_k / k,$ one can show that

\begin{align} H(z) = \exp\left( \sum_{k \geq 1} \gamma_k z^k \right) \text{ and } E(z) = \exp\left(- \sum_{k \geq 1} \gamma_k (-z)^k \right). \end{align}

We have, by using \eqref{eq:forgottenInMonomial}, that

\[ \sum_{\lambda \vdash n} \forgotten_\lambda(\xvec) = \monomial_{1^n}(\xvec), \qquad \sum_{\lambda \vdash n} \monomial_\lambda(\xvec) = \forgotten_{1^n}(\xvec). \]

Newton identities and determinants

We have

\[ r \completeH_r = \sum_{k=1}^r \powerSum_k \completeH_{r-k} \text{ and } r \elementaryE_r = \sum_{k=1}^r (-1)^{k-1} \powerSum_k \elementaryE_{r-k}. \]

The following determinant identities can be useful:

\[ n! \elementaryE_n = \begin{vmatrix} \powerSum_1 & 1 & 0 & \cdots \\ \powerSum_2 & \powerSum_1 & 2 & 0 & \cdots \\ \vdots && \ddots & \ddots \\ \powerSum_{n-1} & \powerSum_{n-2} & \cdots & \powerSum_1 & n-1 \\ \powerSum_n & \powerSum_{n-1} & \cdots & \powerSum_2 & \powerSum_1 \end{vmatrix}. \]

We also have

\[ p_n = \begin{vmatrix} \elementaryE_1 & 1 & 0 & \cdots & \\ 2\elementaryE_2 & \elementaryE_1 & 1 & 0 & \cdots & \\ 3\elementaryE_3 & \elementaryE_2 & \elementaryE_1 & 1 & \cdots & \\ \vdots &&& \ddots & \ddots & \\ n\elementaryE_n & \elementaryE_{n-1} & \cdots & & \elementaryE_1 & \end{vmatrix}. \]

Principal specializations

Given a symmetric function $f,$ we define the following principal specializations:

\[ \princSpec_k^q(f) = f(1,q,q^2,\dotsc,q^{k-1}) \qquad \princSpec_k^1(f) = f(\underbrace{1,1,\dotsc,1}_k) \qquad \princSpec(f) = f(1,q,q^2,\dotsc) \]

The last specialization is a formal power series, and is called the stable principal specialization.

Example (Small example).

For example, consider $\powerSum_1(\xvec)=x_1+x_2+x_3+\dotsb.$ The stable principal specialization is

\[ \powerSum_1(1,q,q^2,\dotsc) = 1+q+q^2+\dotsb = \frac{1}{1-q}. \]

In a similar manner, with $\powerSum_k(\xvec)=x_1^k + x_2^k + x_3^k + \dotsb,$ we find that

\[ \powerSum_k(1,q,q^2,\dotsc) = 1+q^k+q^{2k}+\dotsb = \frac{1}{1-q^k}. \]

Since every symmetric function $f$ can be expressed in terms of power-sum symmetric functions, this gives one possible way for computing $\princSpec(f).$

Let $\lambda$ be a partition of $n$ with $\ell$ parts. Furthermore, let $m_i$ be the number of parts of size $i,$ and we use the notation of raising and falling factorials.

  • $\princSpec_k^1(\monomial_\lambda) = \frac{(k)_{\ell}}{m_1!m_2!\dotsm m_\ell!}$
  • $\princSpec_k^1(\powerSum_\lambda) = k^{\ell}$
  • $\princSpec_k^1(\forgotten_\lambda) = (-1)^{n-\ell}\frac{(k)^{\ell}}{m_1!m_2!\dotsm m_\ell!}$
  • $\princSpec_k^1(\completeH_\lambda) = \prod_{j=1}^\ell \frac{(k)^{\lambda_j}}{\lambda_j!}$
  • $\princSpec_k^1(\elementaryE_\lambda) = \prod_{j=1}^\ell \frac{(k)_{\lambda_j}}{\lambda_j!}$

We also have

\[ \princSpec_k^q( \elementaryE_n ) = q^{\binom{n}{2}} \qbinom{k}{n}_q, \qquad \princSpec_k^q( \completeH_n ) = \qbinom{n+k-1}{n}_q \qquad \princSpec_k^q( \powerSum_n ) = \frac{1-q^{kn}}{1-q^n}. \]

For reference and proofs, see e.g. [Ros02]. Warning! Note that she uses the non-standard notation $|\lambda|=m_1!m_2!\dotsm m_\ell!.$

We also have the following stable principal specializations, see [p. 303, Sta01] for details.

  • $\princSpec(\powerSum_\lambda) = \prod_{j=1}^\ell \frac{1}{1-q^{\lambda_j}} $
  • $\princSpec(\completeH_\lambda) = \prod_{j=1}^\ell \frac{1}{(1-q)(1-q^2)\dotsm (1-q^{\lambda_j})} $
  • $\princSpec(\elementaryE_\lambda) = \prod_{j=1}^\ell \frac{q^{\binom{\lambda_j}{2}}}{(1-q)(1-q^2)\dotsm (1-q^{\lambda_j})} $

Involution on symmetric functions

There is an involution $\omega$ on symmetric functions defined as $\omega(\elementaryE_\lambda) = \completeH_\lambda$ for all $\lambda.$ It acts on the power-sum symmetric functions and the Schur polynomials as

\[ \omega(\powerSum_\lambda) = (-1)^{|\lambda|-\length(\lambda)}\powerSum_\lambda \quad \text{ and } \quad \omega(\schurS_\lambda) = \schurS_{\lambda'}. \]

This involution has natural extensions to quasisymmetric functions, by defining it on the Gessel quasisymmetric functions.

Transition matrices

There are combinatorial interpretations of the coefficients in the transition matrices between these bases. The reference [ER91] gives a comprehensive overview.

In the table below, we show the coefficients in the expansion

\[ \mathrm{v}_\lambda = \sum_{\mu} c_{\lambda \mu} \mathrm{u}_\mu, \]

for different choices of bases $\{\mathrm{v}_\lambda\}$ and $\{\mathrm{u}_\lambda\}.$

$\;$$\completeH_\lambda$$\elementaryE_\lambda$$\schurS_\lambda$$\powerSum_\lambda$$\monomial_\lambda$
$\completeH_\mu$$I$$(-1)^{n-\length(\mu)}|B_{\mu,\lambda}|$$(K_{\lambda,\mu})^{-1}$$(-1)^{\length(\mu)-\length(\lambda)}w(B_{\mu,\lambda})$$(IM_{\lambda,\mu})^{-1}$
$\elementaryE_\mu$$(-1)^{n-\length(\mu)}|B_{\mu,\lambda}|$$I$$(K_{\lambda',\mu})^{-1}$$(-1)^{n-\length(\mu)}w(B_{\mu,\lambda})$$(BM_{\lambda,\mu})^{-1}$
$\schurS_\mu$$K_{\lambda,\mu}$$K_{\lambda',\mu}$$I$$\chi_{\lambda,\mu}$$(K_{\lambda,\mu})^{-1}$
$\powerSum_\mu$$|OB_{\mu,\lambda}|/z_\mu$$(-1)^{n-\length(\mu)}|OB_{\mu,\lambda}|/z_\mu$$\chi_{\lambda,\mu}/z_\mu$$I$$(-1)^{\length(\mu)-\length(\lambda)}w(B_{\lambda,\mu})/z_\mu$
$\monomial_\mu$$IM_{\lambda,\mu}$$BM_{\lambda,\mu}$$K_{\lambda,\mu}$$|OB_{\lambda,\mu}|$$I$

We have that $BM_{\lambda,\mu}$ is the number of $01$-matrices with row sums given by $\lambda$ and column sums given by $\mu.$ Similarly, $IM_{\lambda,\mu}$ is the number of non-negative integer matrices with row sums given by $\lambda$ and column sums given by $\mu.$ Combinatorial interpretations for computing the matrices $(IM_{\lambda,\mu})^{-1}$ and $(BM_{\lambda,\mu})^{-1}$ are given in [KR06].

The coefficient $\chi_{\lambda,\mu}$ is a character value, and can be computed using the Murnaghan–Nakayama rule.

The coefficient $K_{\lambda,\mu}$ is a Kostka coefficient. Moreover, the entries in the inverse Kostka matrix, $(K_{\lambda,\mu})^{-1},$ have a combinatorial formula, see [ER90].

Theorem (Eğecioğlu, Remmel (1990)).

A special rim hook tabloid of shape $\lambda$ and type $\mu$ is a filling of $\lambda$ with rim hooks, such that each rim hook has at least one box in the first column. The sign of a rim hook is $(-1)^{r-1}$ where $r$ is the number of rows it spans, and the sign $\sign(T)$ of a special rim hook tabloid $T$ is the product of the signs of the rim hooks. We have that

\[ (K_{\lambda,\mu})^{-1} = \sum_{T \in \mathrm{RHT}(\lambda,\mu) } \sign(t) \]

where $\mathrm{RHT}(\lambda,\mu)$ is the set of special rim hook tabloids with shape $\lambda$ and type $\mu.$

The coefficients $B_{\mu,\lambda}$ and $OB_{\mu,\lambda}$ may be computed by counting certain brick tabloids, see [ER91].

Positivity probabilities

In [PvW18], the authors describe the probability that a random symmetric function is monomial- schur- and elementary-positive, by comparing volumes of cones spanned by the basis elements.

Hopf algebra

The algebra of symmetric functions admit a Hopf algebra structure, where the coproduct is defined as

\[ \Delta( \elementaryE_k ) = \sum_{j=0}^k \elementaryE_j \otimes \elementaryE_{k-j} \]

See Hopf algebras, symmetric functions and representations, by R. Cheng and Hopf algebras in combinatoris, D. Grinberg and V. Reiner.

References