#Symmetric functions
A function \(f\) in \(n\) variables is symmetric if for any permutation \(w\in \symS_n,\) we have \[f(x_1,x_2,\dotsc,x_n) = f(x_{w(1)},x_{w(2)},\dotsc,x_{w(n)}).\] The space of symmetric functions is denoted \(\spaceSym\). This is a graded vector space, and the Hilbert series of this space is given by the partition numbers, A000041.
Further down we list the most common families of symmetric functions. These are all bases for the space of symmetric functions.
Here are a few useful references:
M. Zabrocki’s introduction to symmetric functions [Zab03].
A. Prasad’s introduction to Schur polynomials, published in [Pra19].
There are also colored analogues attached to wreath products. In [IJS08], F. Ingram, N. Jing, and E. Stitzinger study wreath product Schur functions using colored partitions and tableaux, including analogues of the Pieri and Littlewood–Richardson rules. A. Weinfeld constructs families of bases for symmetric polynomials and uses them to obtain Schur bases for quotient rings generalizing the cohomology and quantum cohomology of Grassmannians [Wei19]. L. G. Fel introduces symmetric polynomials associated with numerical semigroups [Fel21]. These polynomials have power-sum expressions and are conjecturally related to the polynomial part of restricted partition functions.
Cylindric analogues of complete and elementary symmetric functions appear in the work of C. Korff and D. Palazzo [KP20]. They define cylindric symmetric functions using cylindric reverse plane partitions and prove positivity properties for the resulting families.
M. B. Weising introduces Artin symmetric functions [Wei24], attached to Galois representations over number fields. These live in a completed tensor product of symmetric-function rings indexed by prime ideals. The construction is designed so that Artin symmetric functions satisfy induction, inflation, and direct-sum properties analogous to Artin \(L\)-functions, and their arithmetic Hall–Littlewood expansions are described by trace values of representations built from modified Hall–Littlewood functors. M. B. Weising also studies multi-symmetric Schur functions [Wei25]. These are stable limits of key polynomials, form an integral basis for multi-symmetric functions indexed by tuples of partitions, and expand positively into tensor products of ordinary Schur functions.
N. Bonneux, Z. Hamaker, J. Stembridge, and M. Stevens study Wronskians of Appell polynomial families indexed by partitions [BHSS19]. Their recurrence relations, derivative formulas, and integrality results are proved using symmetric-function methods. Milo Bechtloff Weising introduces higher order Bell symmetric functions built from iterated plethystic exponential towers [Wei25]. These symmetric functions are Frobenius characteristics of permutation representations on hyper-partitions, and their monomial and power-sum expansions satisfy explicit recurrences.
#Laurent symmetric functions
A Laurent symmetric polynomial is a symmetric Laurent polynomial, so negative exponents are allowed. H. Fernández, E. Kleinwort, T. Li, and M. Torres-Danta give generators and relations for their ring and prove that Laurent Schur polynomials form a basis [FKLT26, Thm. 4.15]. They extend the Hall inner product so that this basis is orthonormal [FKLT26, Prop. 4.25].
For infinitely many variables, their ring of Laurent symmetric functions is a direct limit of inverse limits [FKLT26, Thm. 5.14]. Its Laurent Schur basis gives the characters of irreducible algebraic representations of \(\GL_\infty\); in finitely many variables, Laurent Schur polynomials are the characters of irreducible rational representations of \(\GL_n.\)
#Polysymmetric functions
Polysymmetric functions form an algebra \(\mathrm{PSym}\) which may be viewed as a scaled tensor product of copies of \(\spaceSym.\) D. Martinez studies these functions using stack partitions [Mar25]. He introduces signed polysymmetric bases and gives transition formulas among monomial, homogeneous, elementary, power, and signed bases, extending familiar symmetric-function identities to the polysymmetric setting.
A. Khanna studies the plethystic bases \(H,\) \(E,\) \(E^+,\) and \(P\) for \(\mathrm{PSym}\) [Kha25]. The transition matrices between these bases have bijective and sign-reversing-involution interpretations, and the paper records several OEIS sequences arising from these transition coefficients.
Vector-space and operator structures on symmetric functions, including Fock space, the Hall inner product, the standard involution, and the Hopf algebra structure, are collected on a separate page.
#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 \(\alpha\) are a rearrangement of the parts of \(\lambda.\)
The augmented monomial symmetric functions are the diagonal rescaling defined by \[\widetilde{\monomial}_\lambda \coloneqq (m_1!m_2!\dotsm ) \monomial_\lambda, \qquad \lambda=(1^{m_1},2^{m_2},\dotsc).\] For example, \(\widetilde{\monomial}_{(2,1,1)}=2\monomial_{(2,1,1)}.\) See [Mer15] for more background.
N. Yamaguchi, Y. Yamaguchi, and G. Shibukawa study principal specializations of monomial symmetric functions at tuples of roots of unity [YYS22]. Their formulas extend Ore’s formulas for the cyclic group determinant and describe coefficients in powers of circulant determinants.
#Elementary symmetric functions
The elementary symmetric functions are defined as follows: \[\elementaryE_k(\xvec) = \sum_{i_1 \lt{} i_2 \lt{} \dotsb \lt{} i_k } x_{i_1} \dotsm x_{i_k} = \monomial_{1^k}(\xvec), \qquad \elementaryE_\lambda(\xvec) = \elementaryE_{\lambda_1} \elementaryE_{\lambda_2} \dotsm \elementaryE_{\lambda_\ell}\] Their generating function is \[\sum_{k\geq 0} \elementaryE_k(\xvec) z^k = \prod_i (1+x_i z).\]
#Complete homogeneous symmetric functions
Analogously to the elementary symmetric functions, the complete homogeneous functions are defined by \[\completeH_k(\xvec) = \sum_{i_1 \leq i_2 \leq \dotsb \leq i_k } x_{i_1} \dotsm x_{i_k} = \sum_{\lambda \vdash k} \monomial_\lambda(\xvec), \qquad \completeH_\lambda(\xvec) = \completeH_{\lambda_1} \completeH_{\lambda_2} \dotsm \completeH_{\lambda_\ell}\] Their generating function is \[\sum_{k\geq 0} \completeH_k(\xvec) z^k = \prod_i \frac{1}{1-x_i z}.\]
A. Heaton and I. Shankar give a sum-of-squares certificate showing that \(\completeH_{(4,4)}-\completeH_{(5,2,1)}\) is nonnegative on the nonnegative orthant [HS21]. This disproves a conjecture that such nonnegativity for complete homogeneous symmetric functions should force dominance comparability, as happens for several other standard bases.
#Power-sum symmetric functions
The power-sum symmetric functions are defined as \[\powerSum_k(\xvec) = x_{1}^k + x_2^k + x_3^k + \dotsb = \monomial_{(k)}(\xvec), \qquad \powerSum_\lambda(\xvec) = \powerSum_{\lambda_1} \powerSum_{\lambda_2} \dotsm \powerSum_{\lambda_\ell}.\] The ordinary generating function for the individual power sums is \[\sum_{k\geq 1} \powerSum_k(\xvec) z^{k-1} = \sum_i \frac{x_i}{1-x_i z}.\]
They serve as an orthogonal basis for the Hall inner product. We have the following product expansions [Mac95, Sec. 1.4] (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 \[\begin{aligned} \prod_{i,j,k\geq 1}(1+x_i y_j z_k) &= \prod_{k\geq 1} \prod_{i,j\geq 1}(1+x_i (y_j z_k)) \\ &= \prod_{k\geq 1} \sum_{\lambda} (-1)^{|\lambda|-\length(\lambda)} z_\lambda^{-1} \powerSum_\lambda(\xvec)\powerSum_\lambda(\yvec z_k). \end{aligned}\] Now using that \(\powerSum_\lambda(\yvec z_k) = z_k^{|\lambda|}\powerSum_\lambda(\yvec),\) we deduce the statement.
There are two quasisymmetric refinements of the power-sum symmetric functions given in [BDHM+20].
Conjecture (Sundaram 2017, [Sun17, Conj. 1]).
Let \(L_n\) be the reverse lexicographic ordering of partitions. Then the following sum over any initial segment, \[\sum_{\lambda \in [1^n,\mu]} \powerSum_\lambda(\xvec)\] is Schur-positive.
For some partial progress on S. Sundaram’s conjecture, see [Sun20].
For a \(q\)-analog of the power-sum symmetric functions, see [Bru23].
#Witt symmetric functions
The Witt symmetric functions form a multiplicative basis \(w_\lambda=w_{\lambda_1}w_{\lambda_2}\dotsm\) of \(\spaceSym\) [IV96]. They may be defined recursively by \[\prod_{d\geq 1}(1-w_d t^d)^{-1} = \sum_{n\geq 0}\completeH_n t^n.\] Equivalently, \[\completeH_n=\sum_{\lambda\vdash n} w_\lambda, \qquad \powerSum_n=\sum_{d\mid n} d\,w_d^{n/d}.\] The name comes from the interpretation of \(\spaceSym\) as the coordinate ring of the group scheme of Witt vectors. In this interpretation the Witt symmetric functions record the Witt components, while the power-sum symmetric functions record the ghost components; see [BW05, Sec. 3.2].
#Forgotten symmetric functions
Finally, the forgotten symmetric functions \(\forgotten_\lambda(\xvec)\) are 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 \(\langle \forgotten_\lambda, \elementaryE_\mu \rangle = \delta_{\lambda \mu}.\)
This basis does not have as many applications as the other standard bases.
R. Stanley gives the following monomial expansion, where \(\lambda \vdash n\) has \(\ell\) parts [Sta01, Ex. 7.9]: \[(-1)^{n-\ell} \forgotten_\lambda(\xvec) = \sum_{\mu} a_{\lambda\mu} \monomial_\mu(\xvec)\] 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
\[\sum_{i=0}^n (-1)^i \elementaryE_i \completeH_{n-i} = \delta_{n,0}.\] Set \[E(z) = \sum_{k \geq 0} \elementaryE_k z^k, \quad P(z) = \sum_{k \geq 0} \powerSum_{k+1} z^k \text{ and } H(z) = \sum_{k \geq 0} \completeH_k z^k\] Then \(E(-z)H(z)=1.\) Moreover, \[P(-z) = \frac{E'(z)}{E(z)}, \quad P(z) = \frac{H'(z)}{H(z)}.\] Also, \[\begin{aligned} H(z) = \exp\left( \sum_{k \geq 1} \frac{\powerSum_k }{k} z^k \right) \text{ and } E(-z) = \exp\left( - \sum_{k \geq 1} \frac{\powerSum_k }{k} z^k \right). \end{aligned}\]
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 \(\schurQ_{(n)}(\xvec).\)
Using the relation above, \[\sum_{\lambda \vdash n} \forgotten_\lambda(\xvec) = \monomial_{1^n}(\xvec), \qquad \sum_{\lambda \vdash n} \monomial_\lambda(\xvec) = \forgotten_{1^n}(\xvec).\]
#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: \[n! \elementaryE_n = \begin{vmatrix} \powerSum_1 & 1 & 0 & \dotsc \\ \powerSum_2 & \powerSum_1 & 2 & 0 & \dotsc \\ \vdots && \ddots & \ddots \\ \powerSum_{n-1} & \powerSum_{n-2} & \dotsc & \powerSum_1 & n-1 \\ \powerSum_n & \powerSum_{n-1} & \dotsc & \powerSum_2 & \powerSum_1 \end{vmatrix}.\]
We also have \[\powerSum_n = \begin{vmatrix} \elementaryE_1 & 1 & 0 & \dotsc & \\ 2\elementaryE_2 & \elementaryE_1 & 1 & 0 & \dotsc & \\ 3\elementaryE_3 & \elementaryE_2 & \elementaryE_1 & 1 & \dotsc & \\ \vdots &&& \ddots & \ddots & \\ n\elementaryE_n & \elementaryE_{n-1} & \dotsc & & \elementaryE_1 & \end{vmatrix}.\]
#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, and let \(m_i\) be the number of parts of size \(i.\) We use the notation of rising 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 M. Rosas uses the non-standard notation \(|\lambda|=m_1!m_2!\dotsm m_\ell!.\)
We also have the following stable principal specializations, see [Sta01, p. 303] for details.
\(\princSpec(\powerSum_\lambda) = \prod_{j=1}^\ell \frac{1}{1-q^{\lambda_j}}\)
\(\princSpec(\completeH_\lambda) = \prod_{j=1}^\ell \frac{1}{(1-q)(1-q^2)\dotsm (1-q^{\lambda_j})}\)
\(\princSpec(\elementaryE_\lambda) = \prod_{j=1}^\ell \frac{q^{\binom{\lambda_j}{2}}}{(1-q)(1-q^2)\dotsm (1-q^{\lambda_j})}\)
C. O’Sullivan studies the combinatorics involving the elementary, power-sum, and complete homogeneous polynomials [O22], where various combinatorial sequences are shown to satisfy similar relations.
B. E. Sagan and J. P. Swanson show that \(q\)-Stirling numbers of type \(B\) can be obtained as the following specialization [SS22]: \[S_B(n,k;q) = \completeH_{n-k}\left([1]_q,[3],[5]_q,\dotsc,[2k+1]_q \right).\] They also prove additional identities.
#Jacobi–Stirling specializations
Fix \(z\gt{}-1.\) The Jacobi–Stirling numbers of the first and second kind, denoted respectively by \(\operatorname{Jc}_n^k(z)\) and \(\operatorname{JS}_n^k(z),\) satisfy \[\begin{aligned} \operatorname{Jc}_n^k(z) &= \operatorname{Jc}_{n-1}^{k-1}(z) +(n-1)(n-1+z)\operatorname{Jc}_{n-1}^k(z), \\ \operatorname{JS}_n^k(z) &= \operatorname{JS}_{n-1}^{k-1}(z) +k(k+z)\operatorname{JS}_{n-1}^k(z). \end{aligned}\] Here both triangles have entry \(1\) at \((0,0),\) and their entries vanish outside \(0\leq k\leq n.\) They are specializations of the elementary and complete homogeneous symmetric functions: \[\begin{aligned} \operatorname{Jc}_n^k(z) &= \elementaryE_{n-k}\bigl(1(1+z),2(2+z),\dotsc,(n-1)(n-1+z)\bigr), \\ \operatorname{JS}_n^k(z) &= \completeH_{n-k}\bigl(1(1+z),2(2+z),\dotsc,k(k+z)\bigr). \end{aligned}\] At \(z=1,\) these become the Legendre–Stirling numbers of the two kinds. The two lower-unitriangular matrices are signed inverses: \[\sum_{k=j}^{i}(-1)^{k+j} \operatorname{JS}_i^k(z)\operatorname{Jc}_k^j(z)=\delta_{i,j}.\] See [CZ23, Secs. 7.5–7.8].
For the first kind, the elementary-symmetric generating identity gives \[\sum_{k=0}^n \operatorname{Jc}_n^k(z)x^k =x\prod_{i=1}^{n-1}\bigl(x+i(i+z)\bigr).\] Thus all its zeros are non-positive. Its coefficients form a Pólya frequency sequence, and Cui–Zhu’s lattice-path construction shows that both the full first-kind triangle and the Toeplitz matrix of every row are totally non-negative [CZ23, Sec. 7.6].
#Transition matrices
There are combinatorial interpretations of the coefficients in the transition matrices between these bases. The reference [ER91] gives a comprehensive overview. I. Pak and C. Robichaux give a systematic discussion of signed combinatorial interpretations for many integral constants in algebraic combinatorics, including structure constants and transition coefficients for standard symmetric and quasisymmetric bases [PR24].
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 \(\{\mathrm{u}_\lambda\}.\)
\( \; \) \( \completeH_\lambda \) \( \elementaryE_\lambda \) \( \schurS_\lambda \) \( \powerSum_\lambda \) \( \monomial_\lambda \) \( \completeH_\mu \) \( I \) \( (-1)^{n-\length(\mu)}|B_{\mu,\lambda}| \) \( (K_{\lambda,\mu})^{-1} \) \( (-1)^{\length(\mu)-\length(\lambda)}w(B_{\mu,\lambda}) \) \( (IM_{\lambda,\mu})^{-1} \) \( \elementaryE_\mu \) \( (-1)^{n-\length(\mu)}|B_{\mu,\lambda}| \) \( I \) \( (K_{\lambda',\mu})^{-1} \) \( (-1)^{n-\length(\mu)}w(B_{\mu,\lambda}) \) \( (BM_{\lambda,\mu})^{-1} \) \( \schurS_\mu \) \( K_{\lambda,\mu} \) \( K_{\lambda',\mu} \) \( I \) \( \chi_{\mu,\lambda} \) \( (K_{\lambda,\mu})^{-1} \) \( \powerSum_\mu \) \( |OB_{\mu,\lambda}|/z_\mu \) \( (-1)^{n-\length(\mu)}|OB_{\mu,\lambda}|/z_\mu \) \( \chi_{\lambda,\mu}/z_\mu \) \( I \) \( (-1)^{\length(\mu)-\length(\lambda)}w(B_{\lambda,\mu})/z_\mu \) \( \monomial_\mu \) \( IM_{\lambda,\mu} \) \( BM_{\lambda,\mu} \) \( K_{\lambda,\mu} \) \( |OB_{\lambda,\mu}| \) \( I \)Here \(BM_{\lambda,\mu}\) 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 nonnegative 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 coefficients \(B_{\mu,\lambda}\) and \(OB_{\mu,\lambda}\) may be computed by counting certain brick tabloids, see [ER91].
#Model for inverse Kostka coefficients
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]. A special rim hook tabloid of type \(\lambda\) and shape \(\mu\) is a filling of \(\mu\) with rim hooks whose lengths are given by \(\lambda,\) and each rim hook has at least one box in the first column. In other words, a special rim hook tabloid is a border strip tableau, where every border strip contains a box from the first column. An important difference with border strip tableaux is that the rim-hooks are not labeled.
The height of a rim hook tableau is defined as for a border strip tableau.
Theorem (Eğecioğlu, Remmel (1990)).
\[(K_{\lambda,\mu})^{-1} = \sum_{T \in \mathrm{SRHT}(\lambda,\mu) } (-1)^{\ht(T)}\] where \(\mathrm{SRHT}(\lambda,\mu)\) is the set of special rim hook tabloids of type \(\lambda\) and shape \(\mu.\)
Example (A special rim hook tableau of type \((8,6,6,4,3,1)\)).
Here is an example of a special rim hook tableau of type \((8,6,6,4,3,1)\) and shape \((7,7,5,3,2,2,2).\)
This tableau is the only contribution to the coefficient \((K_{866431,7753222})^{-1}\) with sign \((-1)^{3}= -1.\) This appears in the expansion \[\schurS_{7753222} = \dotsb + (-1)\completeH_{866431} + \dotsb.\]
Example (The inverse Kostka matrix for \(n=4\)).
The inverse Kostka matrix for \(n=4\) is the \(5\times 5\)-transition matrix below: \[\begin{pmatrix} \schurS_{4} \\ \schurS_{31} \\ \schurS_{22} \\ \schurS_{211} \\ \schurS_{1111} \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 \\ 1 & -1 & -1 & 1 & 0 \\ -1 & 2 & 1 & -3 & 1 \\ \end{pmatrix} \begin{pmatrix} \completeH_{4} \\ \completeH_{31} \\ \completeH_{22} \\ \completeH_{211} \\ \completeH_{1111} \\ \end{pmatrix}.\] The rows and columns are indexed by the partitions of \(4\) in lexicographic order: \((4), (3,1), (2,2), (2,1,1), (1,1,1,1).\) For example, \(\schurS_{31} = (-1)\completeH_{4} + \completeH_{31}.\)
By applying the \(\omega\) involution, we get the transition matrix from the Schur basis to the elementary basis: \[\begin{pmatrix} \schurS_{4} \\ \schurS_{31} \\ \schurS_{22} \\ \schurS_{211} \\ \schurS_{1111} \\ \end{pmatrix} = \begin{pmatrix} -1 & 2 & 1 & -3 & 1 \\ 1 & -1 & -1 & 1 & 0 \\ 0 & -1 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} \elementaryE_{4} \\ \elementaryE_{31} \\ \elementaryE_{22} \\ \elementaryE_{211} \\ \elementaryE_{1111} \\ \end{pmatrix}.\]
This expansion is not cancellation-free in general. For example, \[\schurS_{2211} = \completeH_{42} - \completeH_{51} - \completeH_{222} + \completeH_{411} + \completeH_{2211} - \completeH_{3111}.\] There are two special rim hook tabloids of type \((3,2,1)\) and shape \((2,2,1,1),\) but these have different signs. Hence, the coefficient of \(\completeH_{321}\) in the expansion of \(\schurS_{2211}\) is zero.
#Positivity probabilities
R. Patrias and S. v. Willigenburg describe the probability that a random symmetric function is monomial-, Schur-, and elementary-positive by comparing volumes of cones spanned by the basis elements [PW18].
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