2023-06-29

## 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:

- M. Zabrocki's introduction to symmetric functions
- A. Prasad's introduction to Schur polynomials, published in [Pra19a].

## 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 following product expansions [Sec. 1.4, Mac95] (see the preliminaries for the definition of $z_\lambda$).

\[ \prod_{i,j\geq 1}(1-x_i y_j)^{-1} = \sum_{\lambda} z_\lambda^{-1} \powerSum_\lambda(\xvec)\powerSum_\lambda(\yvec). \]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 [BDHMN20].

**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

\[ E(z) = \sum_{k \geq 0} \elementaryE_k z^k, \quad P(z) = \sum_{k \geq 0} \powerSum_{k+1} z^k \text{ and } H(z) = \sum_{k \geq 0} \completeH_k z^k \]we get $E(-z)H(z)=1.$ Moreover,

\[ P(-t) = \frac{E'(t)}{E(t)}, \quad P(t) = \frac{H'(t)}{H(t)}. \]One can show that

\begin{align} H(z) = \exp\left( \sum_{k \geq 1} \frac{\powerSum_k }{k} z^k \right) \text{ and } E(-z) = \exp\left( - \sum_{k \geq 1} \frac{\powerSum_k }{k} z^k \right). \end{align}Another common identity is

\[ \prod_{i} \frac{1+t x_i}{1-t x_i} = \sum_{\lambda} t^{|\lambda|} 2^{\length(\lambda)} \monomial_{\lambda}(\xvec). \]Note that $\sum_{\lambda \vdash n} 2^{\length(\lambda)} \monomial_{\lambda}(\xvec)$ is the Schur's Q function $\schurQ_{(n)}(\xvec).$

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})} $

The combinatorics involving the elementary, powersum and complete homogeneous polynomials is studied in [O'S22], where various combinatorial sequences are shown to satisfy similar relations.

Sagan–Swanson show in [SS22] that $q$-Stirling numbers of type $B$ can be obtained as the following specialization:

\[ S_B(n,k;q) = \completeH_{n-k}\left([1]_q,[3],[5]_q,\dotsc,[2k+1]_q \right). \]They also show some other interesting identities.

## 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 $\{\mathrm{u}_\lambda\}.$

$\;$ | $\completeH_\lambda$ | $\elementaryE_\lambda$ | $\schurS_\lambda$ | $\powerSum_\lambda$ | $\monomial_\lambda$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$\completeH_\mu$ | $I$ | $(-1)^{n-\length(\mu)}|B_{\mu,\lambda}|$ | $(K_{\lambda,\mu})^{-1}$ | $(-1)^{\length(\mu)-\length(\lambda)}w(B_{\mu,\lambda})$ | $(IM_{\lambda,\mu})^{-1}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$\elementaryE_\mu$ | $(-1)^{n-\length(\mu)}|B_{\mu,\lambda}|$ | $I$ | $(K_{\lambda',\mu})^{-1}$ | $(-1)^{n-\length(\mu)}w(B_{\mu,\lambda})$ | $(BM_{\lambda,\mu})^{-1}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$\schurS_\mu$ | $K_{\lambda,\mu}$ | $K_{\lambda',\mu}$ | $I$ | $\chi_{\mu,\lambda}$ | $(K_{\lambda,\mu})^{-1}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$\powerSum_\mu$ | $|OB_{\mu,\lambda}|/z_\mu$ | $(-1)^{n-\length(\mu)}|OB_{\mu,\lambda}|/z_\mu$ | $\chi_{\lambda,\mu}/z_\mu$ | $I$ | $(-1)^{\length(\mu)-\length(\lambda)}w(B_{\lambda,\mu})/z_\mu$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$\monomial_\mu$ | $IM_{\lambda,\mu}$ | $BM_{\lambda,\mu}$ | $K_{\lambda,\mu}$ | $|OB_{\lambda,\mu}|$ | $I$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

- [BDHMN20] Cristina Ballantine, Zajj Daugherty, Angela Hicks, Sarah Mason and Elizabeth Niese. On quasisymmetric power sums. Journal of Combinatorial Theory, Series A, 175:105273, October 2020.
- [ER90] Ömer Eğecioğlu and Jeffrey B. Remmel. A combinatorial interpretation of the inverse Kostka matrix. Linear and Multilinear Algebra, 26(1-2):59–84, January 1990.
- [ER91] Ömer Eğecioğlu and Jeffrey B. Remmel. Brick tabloids and the connection matrices between bases of symmetric functions. Discrete Applied Mathematics, 34(1-3):107–120, November 1991.
- [KR06] Andrius Kulikauskas and Jeffrey Remmel. Lyndon words and transition matrices between elementary, homogeneous and monomial symmetric functions. The Electronic Journal of Combinatorics, 13(1), February 2006.
- [Mac95] Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications
- [Mer15] Mircea Merca. Augmented monomials in terms of power sums. SpringerPlus, 4(1), November 2015.
- [O'S22] Cormac O'Sullivan. Symmetric functions and a natural framework for combinatorial and number theoretic sequences. arXiv e-prints, 2022.
- [Pra19a] Amritanshu Prasad. An introduction to schur polynomials. The Graduate Journal of Mathematics, 4(2):62–84, 2019.
- [PvW18] Rebecca Patrias and Stephanie van Willigenburg. The probability of positivity in symmetric and quasisymmetric functions. arXiv e-prints, 2018.
- [Ros02] Mercedes H. Rosas. Specializations of MacMahon symmetric functions and the polynomial algebra. Discrete Mathematics, 246(1-3):285–293, March 2002.
- [SS22] Bruce E. Sagan and Joshua P. Swanson. q-Stirling numbers in type B. arXiv e-prints, 2022.
- [Sta01] Richard P. Stanley. Enumerative Combinatorics: Volume 2. Cambridge University Press, First edition, 2001.
- [Sun18] Sheila Sundaram. On a positivity conjecture in the character table of $S_n$. arXiv e-prints, 2018.