2024-03-20

## Schur polynomials

Schur functions were first studied by A.-L. Cauchy in [Cau15], where he defined the Schur functions as a ratio of alternants, and proving that these are symmetric. Later, C. Jacobi [Jac41] gave the definition of Schur polynomials via Cauchy's bialternant formula and proved the Jacobi–Trudi identity. The main application of Schur polynomials in representation theory of the symmetric group and $\GL_n$ came much later when I. Schur studied them in his thesis, [Sch01]. A common problem arising in algebraic combinatorics is to show that some symmetric function is Schur-positive. There are several different techniques for this.

The reader is assumed to be familiar with the most basic families of symmetric functions.

A good overview and proofs of the common formulas involving Schur polynomials
can be found in H. Tamvaki's *The theory of Schur polynomials revisited* [Tam12].

See [CKLMSS19] for a study on the complexity of symbolically computing Schur polynomials.

### Weyl formula

Jacobi's definition of Schur polynomials is via Cauchy's bialternant formula, also called Weyl's formula:

\begin{equation*} \schurS_\lambda(x_1,\dotsc,x_n) = \prod_{1\leq i \lt j \leq n} (x_i-x_j)^{-1} \begin{vmatrix} x_1^{\lambda_1+n-1} & x_2^{\lambda_1+n-1} & \dots & x_n^{\lambda_1+n-1} \\ x_1^{\lambda_2+n-2} & x_2^{\lambda_2+n-2} & \dots & x_n^{\lambda_2+n-2} \\ \vdots & \vdots & \ddots & \vdots \\ x_1^{\lambda_n} & x_2^{\lambda_n} & \dots & x_n^{\lambda_n} \end{vmatrix} \end{equation*}This can also be phrased as

\begin{equation*} \schurS_\lambda(x_1,\dotsc,x_n) = \sum_{\sigma \in \symS_n} \sigma\left(\frac{\xvec^\lambda}{\prod_{i \lt j} 1-x_j/x_i } \right) \end{equation*}where $\sigma$ act by permuting the indices of the variables.

### Combinatorial formula

The Schur polynomials may be defined combinatorially as a sum over semi-standard Young tableaux, see [Mac95Sta01]:

\begin{equation*} \schurS_\lambda(\xvec) = \sum_{T \in \SSYT(\lambda)} x^{T} = \sum_{\mu \vdash n} K_{\lambda\mu}\monomial_\mu(\xvec) \end{equation*}The semi-standard Young tableaux of shape $\lambda$ are in bijection with non-intersecting lattice paths, and Gelfand–Tsetlin patterns. The Kostka coefficients, $K_{\lambda\mu},$ are defined as the number of semistandard Young tableaux with weight $\mu.$

**Example.**

The Schur polynomial $\schurS_{32}(x_1,x_2,x_3)$ is given by

\begin{align*} & x_1^3x_2^2 + x_1^3x_3^2 + x_1^3x_2x_3 + x_1^2x_2^3 + x_1^2x_3^3 +\\ & 2x_1^2x_2x_3^2 + 2x_1^2x_2^2x_3 + x_1x_2x_3^3 + 2x_1x_2^2x_3^2 + x_1x_2^3x_3 + \\ & x_2^2x_3^3 + x_2^3x_3^2 \end{align*}The first few monomials arise from the semi-standard Young tableaux below.

$1$ | $1$ | $1$ |

$2$ | $2$ |

$1$ | $1$ | $1$ |

$3$ | $3$ |

$1$ | $1$ | $1$ |

$2$ | $3$ |

$1$ | $1$ | $2$ |

$2$ | $2$ |

$1$ | $1$ | $3$ |

$3$ | $3$ |

$1$ | $1$ | $2$ |

$3$ | $3$ |

$1$ | $1$ | $3$ |

$2$ | $3$ |

Tokuyama provides an interpolation between the Weyl character formula, and the SSYT definition of Schur functions, by using Gelfand–Tsetlin patterns.

### Specializations

We have that

\[ \schurS_{\lambda}(1^m) = \prod_{1\leq i \lt j \leq m} \frac{\lambda_i-\lambda_j + j-i}{j-i} = \prod_{\square \in \lambda} \frac{m + c(\square)}{h(\square)} \]where $h(\square)$ is the hook of the box, and $c(\square)$ is the content. The first identity is a consequence of the Weyl formula, while the second is the hook-content formula [Sta71].

In fact, we have the $q$-analogue

\[ \schurS_{\lambda}(1,q,q^2,\dotsc,q^{m-1}) = q^{n(\lambda)}\prod_{\square \in \lambda} \frac{1-q^{m + c(\square)}}{1-q^{h(\square)}} \]where $n(\lambda) = \sum_{i}(i-1)\lambda_i.$ In particular,

\[ \schurS_{1^n}(1,q,q^2,\dotsc,q^{m-1}) = q^{n(n-1)/2}\qbinom{m}{n}_q \quad \text{and} \quad \schurS_{(n)}(1,q,q^2,\dotsc,q^{m-1}) = \qbinom{m+n-1}{n}_q. \]For history and proof, see [Kra99].

See also the specializations for the skew Schur functions.

### Hall inner product

Let $\langle \cdot, \cdot \rangle$ be the inner product on symmetric functions such that $\langle \powerSum_\lambda, \powerSum_\mu \rangle = \delta_{\lambda\mu}z_\lambda,$ that is, the power-sum functions form an orthogonal basis. Then $\langle \schurS_\lambda, \schurS_\mu \rangle = \delta_{\lambda\mu},$ so that the Schur polynomials is an orthonormal basis. See the preliminaries for the definition of $z_\lambda.$

### Raising operator formula

The Schur polynomials can be defined as

\[ \schurS_\lambda(\xvec) = \prod_{i \lt j} (1-R_{ij}) \completeH_{\lambda}(\xvec) \]where $R_{ij}$ acts on composition indices $\alpha,$ by increasing the $i^\thsup$ part by 1, and decreasing the $j^\thsup$ part by one. The resulting expression can be recognized as a determinant, namely the first Jacobi–Trudi identity.

This usage of raising operators is not entirely formal, but it can be made precise, see [Gar92].

### Jacobi–Trudi identity

The Jacobi-Trudi identitity expresses the Schur polynomial as a determinant of complete homogeneous symmetric functions:

\begin{equation*} \schurS_\lambda(\xvec) = \det[ \completeH_{\lambda_i + j - i}(\xvec) ]_{1\leq i,j \leq n} \end{equation*}By applying the $\omega$-involution, we recover the dual Jacobi–Trudi identity, also known as the Nägelsbach–Kostka identity:

\begin{equation*} \schurS_{\lambda'}(\xvec) = \det[ \elementaryE_{\lambda_i + j - i}(\xvec) ]_{1\leq i,j \leq n} \end{equation*}To prove these identities, one uses the Lindström–Gessel–Viennot lemma on lattice paths, by bijecting semi-standard Young tableaux to sets of non-intersecting lattice paths.

### Giambelli formula

Let $\lambda=(a_1,\dotsc,a_d|b_1,\dotsc,b_d)$ using Frobenius coordinates. The Giambelli formula states that

\[ \schurS_{\lambda}(\xvec) = \det \left( \schurS_{(a_i|b_j)}(\xvec) \right)_{1\leq i,j \leq d}. \]In other words, it expresses an arbitrary Schur function as a determinant of Schur functions indexed by hook shapes. A combinatorial proof of this formula can be found in [ER88].

A similar result expressing Schur functions as determinants of skew Schur functions indexed by border strips, is proved in [LP88]. This is later generalized in the Hamel–Goulden identities.

### Murnaghan–Nakaygama rule

The Murnaghan–Nakayama rule gives a combinatorial rule for computing the coefficients $\chi^{\lambda}_{\mu}$ in the expansion

\begin{equation*} \schurS_\lambda(\xvec) = \sum_{\mu} \frac{\chi^{\lambda}_{\mu}}{z_\mu} \powerSum_{\mu}(\xvec). \end{equation*}See the page on Murnaghan–Nakaygama rule for more details.

### Hanlon's formula

In [Han88], we have the power-sum expansion

\begin{equation*} H_\lambda \schurS_\lambda(\xvec) = \sum_{\substack{\sigma \in RS(\lambda) \\ \tau \in CS(\lambda)}} \sign(\sigma) \powerSum_{\text{type}(\sigma\tau)}(\xvec) \end{equation*}where $RS(\lambda)$ and $CS(\lambda)$ is the row- and column-stabilizers of a fixed standard Young tableau of shape $\lambda.$

It is conjectured that a variant of Hanlon's formula generalize to Jack polynomials.

### Roichman's formula

In [Roi97], the following power-sum symmetric expansion of a Schur polynomial indexed by $\lambda \vdash n$ is given:

\[ \schurS_\lambda(\xvec) = \sum_{\mu \vdash n} % \sum_{\substack{ T \in \SYT(\lambda) \\ \DES(T) \in U_\mu }} % (-1)^{\DES(T) \setminus S_\mu} \frac{\powerSum_\mu(\xvec)}{z_\mu}. \]Here, $U_\mu$ is the set of $\mu$-unimodal subsets of $[n-1],$ see Athanasiadis formula for the definition of $U_\mu$ and $S_\mu.$

A type-$B$ analogue of this result is give in [AAER17].

### Cauchy identities

The Cauchy identity states that

\begin{equation*} \sum_\lambda \schurS_\lambda(\xvec) \schurS_{\lambda}(y) = \sum_\lambda \monomial_\lambda(\xvec) \completeH_{\lambda}(y)= \prod_{i,j} \frac{1}{1-x_i y_j}. \end{equation*}By applying $\omega$ on the $y$-alphabet, we get the second Cauchy identity

\begin{equation*} \sum_\lambda \schurS_\lambda(\xvec) \schurS_{\lambda'}(y) = \sum_\lambda \monomial_\lambda(\xvec) \elementaryE_{\lambda}(y) = \prod_{i,j} (1+x_i y_j). \end{equation*}One can prove these identities bijectively via the Robinson–Schenstedt–Knuth algorithm.

There are other ways to express the Cauchy identity. For example,

\[ \exp\left( \sum_{m\geq 1} m \frac{\powerSum_m(\xvec)}{m} \frac{\powerSum_m(\yvec)}{m} \right) = \sum_{\lambda} \schurS_\lambda(\xvec)\schurS_\lambda(\yvec). \]See [p. 386, Sta01] for a reference.

In [IT07], the following generalization is proved, where the sum is over all partitions with at most $m$ parts.

\begin{equation*} \sum_{\lambda} t^{\lambda_m} \schurS_\lambda(\xvec_m) \schurS_{\lambda}(\yvec_m) = \prod_{1\leq i,j \leq m} \frac{1}{1-x_i y_j} \frac{1 - \prod_{i=1}^m (x_i y_j) }{ 1 - t \prod_{i=1}^m (x_i y_j) } \end{equation*}See also [BW16], where several nice generalizations of the Cauchy identity are proved. Some of these connect to alternating sign matrices. Another generalization is given in [Bed22].

### Littlewood identities

There are various identities involving a single alphabet,
and are closely related to the theory of *plane partitions*.
Identity two and three below are due to Littlewood.
See [p. 76–77, Mac95] for a reference, or [Exercise 4.3.10, Bre99].

This is the $q=0$ case of Kawanaka's identity.

An integer partition is even if all parts are multiples of two.

\begin{equation*} \sum_{\lambda\text{ even}} \schurS_\lambda(\xvec_m) = \prod_{i=1}^m \frac{1}{1-x^2_i} \prod_{1 \leq i \lt j \leq m} \frac{1}{1-x_ix_j} \end{equation*} \begin{equation*} \sum_{\lambda'\text{ even}} \schurS_\lambda(\xvec_m) = \prod_{1 \leq i \lt j \leq m} \frac{1}{1-x_ix_j} \end{equation*}One can prove these via plethysm, see [Sta01]. Further identities are (see [p. 78–79, Mac95])

\begin{equation*} \sum_{\lambda} (-1)^{n(\lambda)}\schurS_\lambda = \prod_{i}\frac{1}{1-x_i} \prod_{ i \lt j} \frac{1}{1+x_ix_j} \end{equation*}and

\begin{equation*} \sum_{\lambda'\text{ even}} (-1)^{|\lambda|/2}\schurS_\lambda = \prod_{ i \lt j} \frac{1}{1+x_ix_j} \end{equation*}Even more Littlewood-type identities are proved in [Alb21],

### The Kawanaka identity

Kawanaka's identity ([Kaw99]) states that

\begin{equation} \sum_{\mu} \prod_{\square \in \mu} \frac{ 1+q^{h(\square)} }{ 1-q^{h(\square)} } \schurS_\mu = \prod_{i \geq 1} \prod_{r=0}^{\infty} \frac{ 1+x_i q^{r+1} }{ 1- x_i q^{r} } \prod_{i \lt j} \frac{1}{1-x_i x_j}, \end{equation}where hook length of the box, and the first sum is taken over all integer partitions $\mu.$

### Littlewood–Richardson rule

The Littlewood–Richardson rule is a combinatorial rule for multiplying Schur functions:

\[ \schurS_{\lambda} \cdot \schurS_{\mu} = \sum_{\nu} c^{\nu}_{\lambda\mu} \schurS_{\nu}. \]The coefficients $c^{\nu}_{\lambda\mu}$ are the Littlewood–Richardson coefficients.

There are various combinatorial models for the Littlewood–Richardson coefficients.

From a representation-theoretical point of view, let $V^\lambda$ and $V^\mu$ be two irreducible representations of $\symS_n$ and $\symS_m$ respectively. Consider the restriction of the tensor product $V^\lambda \otimes V^\mu$ to $\symS_{m+n}.$ The multiplicity of the irreducible representation $V^\nu$ in this restriction is given by $c^{\nu}_{\lambda\mu}.$

See the stretched Littlewood–Richardson coefficients section for info on the map $n \to c^{n \nu}_{n\lambda,n\mu}.$

### Pieri rule

The Pieri formula is a combinatorial rule for multiplying a Schur function with a complete homogeneous symmetric function:

\[ \completeH_k(\xvec) \schurS_{\lambda}(\xvec) = \sum_{\mu} \schurS_{\mu}. \]The sum is taken over all partitions $\mu \vdash |\lambda|+k,$ obtained by adding $k$ boxes to the diagram $\lambda,$ in such a way that two boxes are not allowed to be added to the same column.

By applying the $\omega$-involution, the *dual Pieri rule* is recovered:

where the sum is now over all partitions $\mu$ obtained by adding $k$ boxes to $\lambda$ in such a way that two added boxes may not appear in the same row.

By iterating the Pieri rules, it follows that $\completeH_\lambda(\xvec)$ and $\elementaryE_\lambda(\xvec)$ are Schur-positive. In fact, it is an easy exercise to see that

\[ \completeH_\lambda(\xvec) = \sum_{\mu} K_{\mu \lambda} \schurS_{\mu}, \]where $K_{\mu \lambda}$ are the Kostka coefficients.

One can view the Pieri rules as special cases of the Littlewood–Richardson rule. An analogue of the Pieri rule for Schubert polynomials is given by Monks rule.

### Kronecker Coefficients

The Kronecker coefficients $g_{\lambda\mu\nu}$ can be defined via the relation

\[ \prod_{i,j,k} \frac{1}{1-x_i y_j z_k} = \sum_{\lambda,\mu,\nu} g_{\lambda\mu\nu} \schurS_\lambda(\xvec)\schurS_\mu(\yvec) \schurS_\nu(\zvec). \]They are related to the characters of the symmetric group, see [pp.68–69, Sch01].

\[ g_{\lambda\mu\nu} =\frac{1}{n!} \sum_{\sigma \in \symS_n} \chi^{\lambda}(\sigma)\chi^{\mu}(\sigma)\chi^{\nu}(\sigma). \]They show up as multiplicities in tensor products of irreducible $\symS_n$-modules, and are therefore non-negative:

\[ S^\mu \otimes S^\mu = \bigoplus_\nu \left( S^\nu \right)^{ \oplus g_{\lambda\mu\nu}} \]In plethystic notation,

\[ \schurS_\nu[XY] = \schurS_\nu[(x_1+x_2+x_3+\dotsb)(y_1+y_2+y_3+\dotsb)] = \sum_{\lambda,\mu} g_{\lambda\mu\nu} \schurS_\lambda(\xvec) \schurS_\lambda(\yvec). \]If define the inner product, or inner tensor product, $\ast,$ on the space of symmetric functions via $\powerSum_\lambda \ast \powerSum_\mu \coloneqq \delta{\lambda \mu} z_\lambda \powerSum_\lambda,$ then

\[ \langle \schurS_\lambda \ast \schurS_\mu, \schurS_\nu \rangle = g_{\lambda\mu\nu}. \]The Kronecker coefficients generalize the Littlewood–Richardson coefficients.

**Theorem (Murnaghan).**

Let $|\nu| = |\lambda|+|\mu|$ and $n>|\nu|.$ Then

\[ g( (n,\nu),\; (n+|\lambda|,\mu),\; (n+|\mu|,\lambda) ) = c^{\nu}_{\lambda \mu}. \]**Problem (Murnaghan 1938).**

It is a major open problem to find a combinatorial interpretation for the Kronecker coefficients.

A combinatorial interpretation for $(g_{\lambda\mu\nu})^2$ is discussed in [GR20],
by using *ribbon graphs*.

**Theorem (Bürgisser and Ikenmeyer, [BI08]).**

Computing $g_{\lambda\mu\nu}$ is $\# P$-hard in general.

**Theorem (Ikenmeyer and Panova, [IP23]).**

Every Kronecker coefficient is a reduced Kronecker coefficient. The reduction can be done in polynomial time, which means that reduced Kronecker coefficients are as hard as computing $g_{\lambda\mu\nu}.$

**Conjecture (Saxl conjecture, 2012).**

Let $\rho$ be the staircase shape $(k-1,k-2,\dotsc,2,1).$ Prove that for all $\nu \vdash \binom{k}{2}$ we have that

\[ g_{\rho, \rho, \nu} \gt 0. \]Equivalently, every $S^{\nu}$ appears in the tensor product $S^\rho \otimes S^\rho.$

This conjecture was presented by Jan Saxl at a UCLA talk.

A very short proof for the tensor cube version of Saxl's conjecture was recently posted at [Ryb22]. This states that for every $\nu \vdash \binom{k}{2},$ we have that $S^{\nu}$ appears in the tensor product $S^\rho \otimes S^\rho \otimes S^\rho.$

## Skew Schur polynomials

The skew Schur polynomials are defined via the property

\[ \langle \schurS_\mu \schurS_\nu, \schurS_{\lambda} \rangle = \langle \schurS_{\lambda/\mu}, \schurS_\nu \rangle. \]In other words, the skew Schur functions may be defined via the Littlewood–Richardson coefficients:

\[ \schurS_\mu \schurS_\nu = \sum_{\lambda} c^{\lambda}_{\mu \nu} \schurS_\lambda \qquad \Leftrightarrow \qquad \schurS_{\lambda/\mu} = \sum_{\nu} c^{\lambda}_{\mu \nu} \schurS_\nu. \]In [GHO20], the authors characterize which skew Schur polynomials are multiplicity-free. That is, for which $\lambda/\mu,$ the coefficients $c^{\lambda}_{\mu \nu}$ are either 0 or 1. Note that they do this for polynomials (symmetric functions are easier).

### Combinatorial formula

The skew Schur polynomials can be expressed as a sum over skew semi-standard Young tableaux:

\begin{equation*} \schurS_{\lambda/\mu}(\xvec) = \sum_{T \in \SSYT(\lambda/\mu)} \xvec^{T}. \end{equation*}In particular, this shows that the family of skew Schur polynomials contain the usual Schur polynomials and is closed under multiplication. That is, $\schurS_{\lambda^1/\mu^1} \schurS_{\lambda^2/\mu^2}$ can be expressed as $ \schurS_{\lambda^3/\mu^3}$ for the skew shape $\lambda^3/\mu^3$ which is the disjoint union of the two diagrams $\lambda^1/\mu^1$ and $\lambda^2/\mu^2.$

### Specializations

Let $\length(\lambda), \length(\mu) \leq n.$ Then

\[ \schurS_{\lambda/\mu}(1,q,q^2,\dotsc) = \left(\prod_{\square \in \lambda/\mu} [n+c(\square)]_q\right)^{-1} \det\left( \qbinom{\lambda_i+n-i}{\lambda_i-\mu_j-i+j} \right)_{1 \leq i,j \leq n}, \]see [Exer. 102, Sta01] and [CS16].

### Jacobi–Trudi identity

Suppose $\lambda/\mu$ is a skew shape with at most $\ell$ rows. Then

\[ \schurS_{\lambda/\mu} = \det\left[ \completeH_{\lambda_i-\mu_j - i + j} \right]_{1\leq i,j \leq \ell} \text{ and } \schurS_{\lambda'/\mu'} = \det\left[ \elementaryE_{\lambda_i-\mu_j - i + j} \right]_{1\leq i,j \leq \ell}. \]This is proved using the Lindström–Gessel–Viennot lemma.

### Murnaghan–Nakaygama rule

Using the same notation as for the Schur polynomials,

\begin{equation*} \schurS_{\lambda/\mu}(\xvec) = \sum_{\nu} \frac{\powerSum_{\nu}(\xvec)}{z_\nu} \sum_{T \in \BST(\lambda/\mu,\nu)} (-1)^{\mathrm{ht}(T)} \end{equation*}where the sum is taken over border strip tableaux of skew shape $\lambda/\mu$ and weight $\nu.$

### Hamel–Goulden identities

In [HG95], A.M Hamel and I.P Goulden describes a uniform framework, unifying the Jacobi–Trudi identities, the Giambelli formula and the result by A. Lascoux and P. Pragacz [LP88].

Given any *outside decompositon* $(\theta_1,\dotsc,\theta_m)$ of $\lambda/\mu,$
one has

See the reference [HG95] for exact definitions and notation.

### Fundamental quasi-symmetric expansion

Skew Schur polynomials and thus Schur polynomials, expand positively in the Gessel fundamental quasi-symmetric functions,

\[ \schurS_{\lambda/\mu}(\xvec) = \sum_{T \in \SYT(\lambda/\mu)} \gessel_{D(T)}(\xvec) \]where $\SYT(\lambda/\mu)$ is the set of standard Young tableaux of shape $\lambda/\mu$ and $D(T)$ is the descent set of $T.$

The slinky rule can be used to find the Schur expansion of a symmetric function, given the fundamental quasi-symmetric expansion.

### Littlewood-type identities

See [Mac95] for the following identity:

\begin{equation*} \sum_{\lambda} \schurS_{\lambda/\mu} = \prod_{i=1} \frac{1}{1-x_i} \prod_{1 \leq i \lt j \leq m} \frac{1}{1-x_ix_j} \sum_{\nu} \schurS_{\mu/\nu}. \end{equation*}## Various topics

### Gustavsson–Milne formula

Let $\lambda$ be a partition with $k$ parts. Then

\[ \schurS_{\lambda_1 - n + 1, \lambda_2 - n + 2,\dotsc,\lambda_k - n + k}(x_1,\dotsc,x_k) =\sum_{S \in \binom{[n]}{k}} \frac{\schurS_\lambda(x_S)}{\prod_{i\in S} \prod_{j\in \overline{S}}} (x_i-x_j). \]For a proof using words such as Chern classes, tautological bundles and Atiyah–Bott localization theorem, see [Thm. 7.1, FRN12].

This identity is closely related to Widom's formula and stretched Schur polynomials, see [Ale12bAle14].

### Cumulants of Schur polynomials

Maciej Dołęga has introduced the notion of cumulants of families of symmetric functions [Doł19]. Let $\{X_\lambda\}$ be such a family, indexed by partitions. The cumulant is defined as

\[ \kappa( X_{\lambda^1}, \dotsc, X_{\lambda^k} ) \coloneqq \sum_{\pi \in \mathrm{SetPart}([r])} (-1)^{|\pi|-1}(|\pi|-1)! \prod_{ B \in \pi } X_{\lambda^{B_1} \oplus \dotsb \oplus \lambda^{B_\ell}} \]where $\mathrm{SetPart}([r])$ denotes set partitions of $[r],$ $|\pi|$ denotes the number of blocks of $\pi$ and $\oplus$ denotes entrywise addition of the parts of the partitions.

**Problem (M. Dołęga, 2017+).**

Prove that $\kappa( X_{\lambda^1}, \dotsc, X_{\lambda^k} )$ is Schur-positive.

There is a generalization of this to modified Macdonald polynomials.

## References

- [AAER17] Ron M. Adin, Christos A. Athanasiadis, Sergi Elizalde and Yuval Roichman. Character formulas and descents for the hyperoctahedral group. Advances in Applied Mathematics, 87:128–169, June 2017.
- [Alb21] Seamus P. Albion. Proof of some Littlewood identities conjectured by Lee, Rains and Warnaar. arXiv e-prints, 2021.
- [Ale12b] Per Alexandersson. Schur polynomials, banded Toeplitz matrices and Widom's formula. Electronic Journal of Combinatorics, 19(4):1–13, 2012.
- [Ale14] Per Alexandersson. Stretched skew Schur polynomials are recurrent. Journal of combinatorial theory. Series A, 122:1–8, 2014.
- [Bed22] Leonid Bedratyuk. Some new identities for Schur polynomials. arXiv e-prints, 2022.
- [BI08] Peter Bürgisser and Christian Ikenmeyer. The complexity of computing Kronecker coefficients. 20th International Conference on Formal Power Series and Algebraic Combinatorics. Discrete Mathematics and Theoretical Computer Science, 2008.
- [Bre99] David Bressoud. Proofs and confirmations: the story of the alternating sign matrix conjecture. Cambridge University Press, 1999.
- [BW16] D. Betea and M. Wheeler. Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices. Journal of Combinatorial Theory, Series A, 137:126–165, January 2016.
- [Cau15] Augustin Louis Cauchy. Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions opérées entre les variables qu'elles renferment. Journal de l'École polytechnique, 10(17):29–112, 1815. Œuvres, ser. 2, vol. 1, pp. 91–169
- [CKLMSS19] Prasad Chaugule, Mrinal Kumar, Nutan Limaye, Chandra Kanta Mohapatra, Adrian She and Srikanth Srinivasan. Schur polynomials do not have small formulas if the determinant doesn't!. arXiv e-prints, 2019.
- [CS16] Xiaomei Chen and Richard P. Stanley. A formula for the specialization of skew Schur functions. Annals of Combinatorics, 20(3):539–548, April 2016.
- [Doł19] Maciej Dołęga. Macdonald cumulants, G-inversion polynomials and G-parking functions. European Journal of Combinatorics, 75:172–194, January 2019.
- [ER88] Ömer N. Eğecioğlu and Jeffrey B. Remmel. A combinatorial proof of the Giambelli identity for Schur functions. Advances in Mathematics, 70(1):59–86, July 1988.
- [FRN12] László M. Fehér, Richárd Rimányi and András Némethi. Equivariant classes of matrix matroid varieties. , 2012.
- [Gar92] Adriano M. Garsia. Orthogonality of Milne's polynomials and raising operators. Discrete Mathematics, 99(1-3):247–264, April 1992.
- [GHO20] Shiliang Gao, Reuven Hodges and Gidon Orelowitz. Multiplicity-free skew Schur polynomials. arXiv e-prints, 2020.
- [GR20] Joseph Ben Geloun and Sanjaye Ramgoolam. Quantum mechanics of bipartite ribbon graphs: integrality, lattices and Kronecker coefficients. arXiv e-prints, 2020.
- [Han88] Phil Hanlon. Jack symmetric functions and some combinatorial properties of Young symmetrizers. Journal of Combinatorial Theory, Series A, 47(1):37–70, January 1988.
- [HG95] A. M. Hamel and I. P. Goulden. Planar decompositions of tableaux and Schur function determinants. European Journal of Combinatorics, 16(5):461–477, September 1995.
- [IP23] Christian Ikenmeyer and Greta Panova. All Kronecker coefficients are reduced Kronecker coefficients. arXiv e-prints, 2023.
- [IT07] Masao Ishikawa and Hiroyuki Tagawa. Schur function identities and hook length posets. 19th International Conference on Formal Power Series and Algebraic Combinatorics. 2007.
- [Jac41] C.G.J. Jacobi. De functionibus alternantibus earumque divisione per productum e differentiis elementorum conflatum. Journal für die reine und angewandte Mathematik, 22:360–371, 1841.
- [Kaw99] Noriaki Kawanaka. A $q$-series identity involving Schur functions and related topics. Osaka Journal of Mathematics, 36(1):157 – 176, 1999.
- [Kra99] Christian Krattenthaler. Another involution principle-free bijective proof of Stanley's hook-content formula. Journal of Combinatorial Theory, Series A, 88(1):66–92, October 1999.
- [LP88] Alain Lascoux and Piotr Pragacz. Ribbon Schur functions. European Journal of Combinatorics, 9(6):561–574, November 1988.
- [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
- [Roi97] Yuval Roichman. A recursive rule for Kazhdan–Lusztig characters. Advances in Mathematics, 129(1):25–45, July 1997.
- [Ryb22] Christopher Ryba. A tensor-cube version of the Saxl conjecture. arXiv e-prints, 2022.
- [Sch01] Issai Schur. Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen. Dieterich, 1901.
- [Sta01] Richard P. Stanley. Enumerative Combinatorics: Volume 2. Cambridge University Press, First edition, 2001.
- [Sta71] Richard P. Stanley. Theory and application of plane partitions. Part 2. Studies in Applied Mathematics, 50(3):259–279, September 1971.
- [Tam12] Harry Tamvakis. The theory of Schur polynomials revisited. L'Enseignement Mathématique, 58(1):147–163, 2012.