#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 proved 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. Tamvakis’ The theory of Schur polynomials revisited [Tam12]. N. Jing and N. Rozhkovskaya give generating functions for shifted symmetric functions [JR19].
For the complexity of symbolically computing Schur polynomials, see [CKLM+23].
#Weyl formula
Jacobi’s definition of Schur polynomials is via Cauchy’s bialternant formula, also called Weyl’s formula: \[\schurS_\lambda(x_1,\dotsc,x_n) = \prod_{1\leq i \lt j \leq n} (x_i-x_j)^{-1} \begin{vmatrix} x_1^{\lambda_1+n-1} & x_2^{\lambda_1+n-1} & \dotsc & x_n^{\lambda_1+n-1} \\ x_1^{\lambda_2+n-2} & x_2^{\lambda_2+n-2} & \dotsc & x_n^{\lambda_2+n-2} \\ \vdots & \vdots & \ddots & \vdots \\ x_1^{\lambda_n} & x_2^{\lambda_n} & \dotsc & x_n^{\lambda_n} \end{vmatrix}\]
Equivalently, \[\schurS_\lambda(x_1,\dotsc,x_n) = \sum_{\sigma \in \symS_n} \sigma\left(\frac{\xvec^\lambda}{\prod_{i \lt j} 1-x_j/x_i } \right)\] where \(\sigma\) acts by permuting the indices of the variables.
#Combinatorial formula
The Schur polynomials may be defined combinatorially as a sum over semistandard Young tableaux, see [Mac95, Sta01]: \[\schurS_\lambda(\xvec) = \sum_{T \in \SSYT(\lambda)} x^{T} = \sum_{\mu \vdash n} K_{\lambda\mu}\monomial_\mu(\xvec)\] The semistandard 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 \[\begin{aligned} & x_1^3x_2^2 + x_1^3x_3^2 + x_1^3x_2x_3 + x_1^2x_2^3 + x_1^2x_3^3 +\\ & 2x_1^2x_2x_3^2 + 2x_1^2x_2^2x_3 + x_1x_2x_3^3 + 2x_1x_2^2x_3^2 + x_1x_2^3x_3 + \\ & x_2^2x_3^3 + x_2^3x_3^2 \end{aligned}\] The first few monomials arise from the semistandard Young tableaux below.
Tokuyama provides an interpolation between the Weyl character formula, and the SSYT definition of Schur functions, by using Gelfand–Tsetlin patterns.
#Specializations
The specialization formula is \[\schurS_{\lambda}(1^m) = \prod_{1\leq i \lt j \leq m} \frac{\lambda_i-\lambda_j + j-i}{j-i} = \prod_{\square \in \lambda} \frac{m + c(\square)}{h(\square)}\] where \(h(\square)\) is the hook length 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].
There is also 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].
M. Hidaka and M. Itoh study Schur polynomials evaluated at all primitive \(n\)th roots of unity [HI24]. They prove that these values are always \(1,\) \(0,\) or \(-1\) when \(n\) has at most two distinct odd prime factors. Their proof reduces the statement to unimodularity of tensor products of certain vector configurations, generalizing familiar restrictions on coefficients of cyclotomic polynomials and their inverses.
G. Panova and L. Petrov give two proofs of a multivariate skew hook-length formula for skew shapes [PP26]. One proof uses contour integrals, while the other interprets excited diagrams as a free-fermion vertex model and derives the identity from the Yang–Baxter equation. This gives another bridge between skew Schur specializations, excited diagrams, non-intersecting lattice paths, and integrable lattice models.
See also the specializations for the skew Schur functions.
#Schur generating functions
A Schur generating function is a normalized generating series in Schur polynomials attached to a probability measure on signatures. Such generating functions are useful because analytic information about the Schur series can be translated into limit-shape and fluctuation statements.
A. Bufetov, L. Petrov, and P. Zografos use Schur generating functions to study random domino tilings of the Aztec diamond in a random environment [BPZ25]. Their results extend law-of-large-numbers and central-limit-theorem tools for particle systems via Schur generating functions and identify different fluctuation regimes for the domino height function.
#Elliptic Schur functions
Elliptic Schur functions are elliptic analogues of Schur functions obtained by replacing the ordinary lattice-path or tableau weights by theta-function weights. M. Schlosser gives such a construction from elliptically weighted non-intersecting lattice paths [Sch07]. The resulting functions have a closed product formula under principal specialization, but they are not known to be orthogonal with respect to a natural elliptic scalar product.
K. Motegi derives a Cauchy formula for elliptic Schur functions from scalar products in the elliptic Felderhof model [Mot18]. This places elliptic Schur functions closer to elliptic hypergeometric and integrable-model constructions than to the ordinary bases of \(\spaceSym.\)
#Hall inner product
The Schur functions form an orthonormal basis under the Hall inner product.
#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 can be made precise; see [Gar92].
#Jacobi–Trudi identity
The Jacobi–Trudi identity expresses the Schur polynomial as a determinant of complete homogeneous symmetric functions: \[\schurS_\lambda(\xvec) = \det[ \completeH_{\lambda_i + j - i}(\xvec) ]_{1\leq i,j \leq n}\] Applying the \(\omega\)-involution gives the dual Jacobi–Trudi identity, also known as the Nägelsbach–Kostka identity : \[\schurS_{\lambda'}(\xvec) = \det[ \elementaryE_{\lambda_i + j - i}(\xvec) ]_{1\leq i,j \leq n}\] J. Blasiak, Omesh Dhar Dwivedi, and D. Grinberg study Jacobi–Trudi determinants over finite fields [BDG23]. They compute vanishing probabilities for families of Jacobi–Trudi and skew Jacobi–Trudi determinants, including arithmetic-progression shapes and ribbon skew shapes. T. Gui and A. L. B. Yang give a proof of the classical Jacobi–Trudi identities for skew Schur polynomials from the BGG category \(\mathcal{O}\) and the Weyl character formula [GY22]. They also interpret Schur coefficients of certain Jacobi–Trudi truncations as tensor product multiplicities in category \(\mathcal{O}.\) S. V. Sam and J. Weyman explain Gessel’s Jacobi–Trudi-like determinant through a construction of A. Zelevinsky involving BGG complexes [SW23]. This gives common ground for the classical Jacobi–Trudi identity and several generalizations related to determinantal varieties. J. S. Kim and M. Yoo use the Bazin identity to obtain determinant formulas for generalized Schur functions [KY21]. M. Fulmek and M. Kleber give bijective proofs for Schur-function identities implying Dodgson condensation and Plücker relations [FK01]. To prove these identities, one uses the Lindström–Gessel–Viennot lemma on lattice paths, by bijecting semistandard 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 [NER88].
A similar result, expressing Schur functions as determinants of skew Schur functions indexed by border strips, is proved in [LP88]. This is later generalized by the Hamel–Goulden identities.
#Murnaghan–Nakayama rule
The Murnaghan–Nakayama rule gives a combinatorial rule for computing the coefficients \(\chi^{\lambda}_{\mu}\) in the expansion \[\schurS_\lambda(\xvec) = \sum_{\mu} \frac{\chi^{\lambda}_{\mu}}{z_\mu} \powerSum_{\mu}(\xvec).\]
See the page on Murnaghan–Nakayama rule for more details.
#Hanlon’s formula
In [Han88], we have the power-sum expansion \[H_\lambda \schurS_\lambda(\xvec) = \sum_{\substack{\sigma \in RS(\lambda) \\ \tau \in CS(\lambda)}} \sign(\sigma) \powerSum_{\mathrm{type}(\sigma\tau)}(\xvec)\] where \(RS(\lambda)\) and \(CS(\lambda)\) are the row- and column-stabilizers of a fixed standard Young tableau of shape \(\lambda.\)
It is conjectured that a variant of Hanlon’s formula generalizes to Jack polynomials.
V. Féray gives a representation-theory-free proof of the Schur-function case in his FPSAC 2015 slides [Fe15].
#Roichman’s formula
Y. Roichman gives the following power-sum expansion of a Schur polynomial indexed by \(\lambda \vdash n\) [Roi97]: \[\schurS_\lambda(\xvec) = \sum_{\mu \vdash n} % \sum_{\substack{ T \in \SYT(\lambda) \\ \DES(T) \in U_\mu }} % (-1)^{\DES(T) \setminus S_\mu} \frac{\powerSum_\mu(\xvec)}{z_\mu}.\] 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 given in [AAER17].
#Cauchy identities
The Cauchy identity states that \[\sum_\lambda \schurS_\lambda(\xvec) \schurS_{\lambda}(y) = \sum_\lambda \monomial_\lambda(\xvec) \completeH_{\lambda}(y) = \prod_{i,j} \frac{1}{1-x_i y_j}.\] Applying \(\omega\) on the \(y\)-alphabet gives the second Cauchy identity \[\sum_\lambda \schurS_\lambda(\xvec) \schurS_{\lambda'}(y) = \sum_\lambda \monomial_\lambda(\xvec) \elementaryE_{\lambda}(y) = \prod_{i,j} (1+x_i y_j).\]
These identities can be proved bijectively via the Robinson–Schensted–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 [Sta01, p. 386] for a reference.
In [IT07], the following generalization is proved, where the sum is over all partitions with at most \(m\) parts. \[\sum_{\lambda} t^{\lambda_m} \schurS_\lambda(\xvec_m) \schurS_{\lambda}(\yvec_m) = \prod_{1\leq i,j \leq m} \frac{1}{1-x_i y_j} \frac{1 - \prod_{i=1}^m (x_i y_j) }{ 1 - t \prod_{i=1}^m (x_i y_j) }\]
See also [BW16], where refined Cauchy and Littlewood identities are proved and related to plane partitions and symmetry classes of alternating sign matrices. Another generalization is given by L. Bedratyuk [Bed25]. Let \(f_0,f_1,\dotsc\) be a family of polynomials in \(\yvec,\) set \(f_r=0\) for \(r\lt 0,\) and write \[F(\yvec,z)\coloneqq \sum_{r\geq 0} f_r z^r.\] Then \[\sum_{\lambda} \schurS_\lambda(\xvec)\det\left[f_{\lambda_i-i+j}\right]_{i,j=1}^n = \prod_{i=1}^n F(\yvec,x_i),\] where the sum is over partitions \(\lambda\) with at most \(n\) parts. Taking \(f_r=\completeH_r(\yvec)\) recovers the ordinary Cauchy identity by the Jacobi–Trudi identity, and the paper also proves bounded variants.
A. Mironov and A. Morozov study averaging procedures for Cauchy identities in symmetric functions [MM23]. Their approach averages over power-sum variables and gives a uniform viewpoint on several previously separate identities. P. Díaz and E. Mainar use Schur functions to study total positivity of collocation matrices of analytic bases [DM26]. Their formulas for initial minors give sufficient conditions for total positivity and lead to Cauchy-identity generalizations for certain analytic function families. D. Stanton derives the Rogers–Ramanujan identities from the Cauchy identity for Schur functions [Sta25]. This gives a short symmetric-function route to the classical product-sum identities. M. Tierz gives a rectangular-Schur formula for a banded Toeplitz determinant and uses the dual Cauchy identity to relate another Toeplitz determinant to rectangular domino tilings. He derives exact finite-size corrections comparing the two determinants [Tie26].
#Littlewood identities
There are various identities involving a single alphabet, closely related to the theory of plane partitions. The second and third identities below are due to D. Littlewood. See [Mac95, p. 76–77] for a reference, or [Bre99, Exercise 4.3.10]. \[\sum_{\lambda} \schurS_\lambda(\xvec_m) = \prod_{i=1}^m \frac{1}{1-x_i} \prod_{1 \leq i \lt j \leq m} \frac{1}{1-x_ix_j}\] This is the \(q=0\) case of Kawanaka’s identity.
An integer partition is even if all parts are multiples of two. \[\sum_{\lambda\text{ even}} \schurS_\lambda(\xvec_m) = \prod_{i=1}^m \frac{1}{1-x^2_i} \prod_{1 \leq i \lt j \leq m} \frac{1}{1-x_ix_j}\]
\[\sum_{\lambda'\text{ even}} \schurS_\lambda(\xvec_m) = \prod_{1 \leq i \lt j \leq m} \frac{1}{1-x_ix_j}\]
These can be proved via plethysm; see [Sta01]. Further identities are given in [Mac95, p. 78–79]: \[\sum_{\lambda} (-1)^{n(\lambda)}\schurS_\lambda = \prod_{i}\frac{1}{1-x_i} \prod_{ i \lt j} \frac{1}{1+x_ix_j}\] and \[\sum_{\lambda'\text{ even}} (-1)^{|\lambda|/2}\schurS_\lambda = \prod_{ i \lt j} \frac{1}{1+x_ix_j}\]
Even more Littlewood-type identities are proved in [Alb21].
F. Schreier-Aigner gives growth-diagram proofs for families of Littlewood identities [Sch24]. The approach introduces projection identities for up and down operators, giving RSK-like bijections which generalize the classical growth-diagram proofs of the Cauchy and dual Cauchy identities.
J. Huh, J. S. Kim, C. Krattenthaler, and S. Okada study bounded Littlewood identities in which the number of odd rows or odd columns is fixed [HKKO25]. Their formulas use determinants and skewing operators, and give consequences for standard Young tableaux, up-down tableaux, and vacillating tableaux with restricted shapes.
I. Fischer proves a bounded version of a generalized Littlewood identity related to alternating sign matrices [Fis24]. The identity appears in the study of alternating sign triangles and trapezoids, and the paper gives combinatorial interpretations of both sides. I. Fischer and H. Höngesberg prove a Littlewood-type identity for modified Robbins polynomials [FH25]. These polynomials generalize Schur polynomials and enumerate down-arrowed monotone triangles; the product side involves a Pfaffian interpreted through six-vertex configurations for diagonally symmetric alternating sign matrices.
Z. Cai, B. Jiang, N. Jing, Z. Li, and Q. Ye give vertex-operator proofs of the dual Littlewood identities in types \(B,\) \(C,\) and \(D\) [CJJL+26]. Their generalized identities sum over restricted or \((-n)\)-asymmetric partitions and yield determinantal measures on partitions.
#The Kawanaka identity
Kawanaka’s identity ([Kaw99]) states that \[\sum_{\mu} \prod_{\square \in \mu} \frac{ 1+q^{h(\square)} }{ 1-q^{h(\square)} } \schurS_\mu = \prod_{i \geq 1} \prod_{r=0}^{\infty} \frac{ 1+x_i q^{r+1} }{ 1- x_i q^{r} } \prod_{i \lt j} \frac{1}{1-x_i x_j},\] where \(h(\square)\) is the 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 information 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.
Applying the \(\omega\)-involution gives the dual Pieri rule: \[\elementaryE_k(\xvec) \schurS_{\lambda}(\xvec) = \sum_{\mu} \schurS_{\mu},\] 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.
Iterating the Pieri rules shows that \(\completeH_\lambda(\xvec)\) and \(\elementaryE_\lambda(\xvec)\) are Schur-positive. Indeed, \[\completeH_\lambda(\xvec) = \sum_{\mu} K_{\mu \lambda} \schurS_{\mu},\] where \(K_{\mu \lambda}\) are the Kostka coefficients.
The Pieri rules are special cases of the Littlewood–Richardson rule. An analogue of the Pieri rule for Schubert polynomials is given by Monk’s 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 [Sch01, pp. 68–69]. \[g_{\lambda\mu\nu} =\frac{1}{n!} \sum_{\sigma \in \symS_n} \chi^{\lambda}(\sigma)\chi^{\mu}(\sigma)\chi^{\nu}(\sigma).\]
J. Blasiak gives a positive combinatorial formula for the hook-shape case [Bla12]. If \(\mu(d)=(n-d,1^d),\) then \(g_{\lambda,\mu(d),\nu}\) counts colored Yamanouchi tableaux of content \(\lambda,\) total color \(d,\) and shape \(\nu\) whose southwest corner is unbarred. The construction uses M. Haiman’s mixed insertion for colored words.
They appear as multiplicities in tensor products of irreducible \(\symS_n\)-modules, and are therefore nonnegative: \[S^\lambda \otimes S^\mu = \bigoplus_\nu \left( S^\nu \right)^{ \oplus g_{\lambda\mu\nu}}\] In plethystic notation, \[\schurS_\nu[XY] = \schurS_\nu[(x_1+x_2+x_3+\dotsb)(y_1+y_2+y_3+\dotsb)] = \sum_{\lambda,\mu} g_{\lambda\mu\nu} \schurS_\lambda(\xvec) \schurS_\mu(\yvec).\]
If we 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 \(n\gt{}|\nu|.\) 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 \(g_{\lambda\mu\nu}.\)
T. Gui formulates conjectures on reduced Kronecker coefficients [Gui22]. These are stable tensor-product conjectures for symmetric group representations, generalizing both A. Okounkov’s log-concavity conjecture for Littlewood–Richardson coefficients and the Schur log-concavity theorem of T. Lam, A. Postnikov, and P. Pylyavskyy.
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 \[g_{\rho, \rho, \nu} \gt 0.\] Equivalently, every \(S^{\nu}\) appears in the tensor product \(S^\rho \otimes S^\rho.\)
This conjecture was presented by J. Saxl at a UCLA talk. A September 2026 preprint by OpenAI announces a proof for every staircase size [Ope26, Thm. 1.1]. The repository also provides a Lean formalization.
C. Bessenrodt and C. Bowman study symmetric and alternating parts of Kronecker squares, with applications to the Saxl conjecture [BB23]. Their results connect these splittings with \(2\)-decomposition numbers and Catalan combinatorics.
A very short proof for the tensor-cube version of Saxl’s conjecture appears in [Ryb22]. This states that for every \(\nu \vdash \binom{k}{2},\) \(S^{\nu}\) appears in the tensor product \(S^\rho \otimes S^\rho \otimes S^\rho.\)
If \(\lambda^\vee\) is the complement of \(\lambda\) in a fixed rectangle, R. Hodges and H. Yin prove Kleber’s conjecture: the products \(\schurS_\lambda\schurS_{\lambda^\vee},\) indexed by unordered complementary pairs, are linearly independent over every commutative ring [HY26]. More generally, for a fixed partition \(\theta,\) the products \(\schurS_\alpha\schurS_\beta\) are linearly independent as \(\{\alpha,\beta\}\) ranges over the componentwise splittings \(\alpha+\beta=\theta.\) This also gives the corresponding independence for Koike–Terada universal-character products.
#Skew Schur polynomials
The skew Schur polynomials are defined via the property \[\langle \schurS_\mu \schurS_\nu, \schurS_{\lambda} \rangle = \langle \schurS_{\lambda/\mu}, \schurS_\nu \rangle.\] We may view multiplication by \(\schurS_\mu\) as a linear map on the space of symmetric functions. This map has an adjoint under the Hall inner product. The adjoint, which is called skewing by \(\mu\) produces \(\schurS_{\lambda/\mu}\) when acting on \(\schurS_{\lambda}.\)
In other words, the skew Schur functions may be defined via the Littlewood–Richardson coefficients: \[\schurS_\mu \schurS_\nu = \sum_{\lambda} c^{\lambda}_{\mu \nu} \schurS_\lambda \qquad \iff \qquad \schurS_{\lambda/\mu} = \sum_{\nu} c^{\lambda}_{\mu \nu} \schurS_\nu.\]
The flagged skew Schur polynomials are nonsymmetric row-bounded refinements of these functions. They specialize back to \(\schurS_{\lambda/\mu}(x_1,\dotsc,x_n)\) when every row is allowed to use all variables \(x_1,\dotsc,x_n.\)
Every finite-variable skew Schur polynomial is also obtained by setting extra variables equal to zero in a Schubert polynomial indexed by a stabilized \(321\)-avoiding permutation [BJS93, Zha26]. This realization is the key input in the proof that the normalization of \(\schurS_{\lambda/\mu}(x_1,\dotsc,x_n)\) is Lorentzian.
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 \(1.\) They do this for polynomials; the symmetric-function case is easier.
P. McNamara and S. v. Willigenburg introduce a diagram operation producing skew shapes with equal skew Schur functions [MW09]. The operation unifies many earlier coincidences and suggests a conjectural classification of skew diagrams with identical skew Schur functions.
E. Y. Jin and S. X. Li study equivalence classes of thickened ribbon Schur functions [JL24]. They characterize connected skew diagrams with exactly one \(2\times m\) or \(m\times 2\) block, showing that the possible equivalence-class sizes in this setting are one, two, or four.
D. Grinberg, N. Korniichuk, K. Molokanov, and S. Khomych give an explicit formula for the diagonal derivative \(\partial/\partial x_1+\dotsb+\partial/\partial x_n\) applied to a skew Schur polynomial [GKMK24]. This generalizes A. Weigandt’s formula for ordinary Schur polynomials.
#Combinatorial formula
The skew Schur polynomials can be expressed as a sum over skew semistandard Young tableaux: \[\schurS_{\lambda/\mu}(\xvec) = \sum_{T \in \SSYT(\lambda/\mu)} \xvec^{T}.\] In particular, this shows that the family of skew Schur polynomials contains 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 [Sta01, Exer. 102] and [CS16].
For a fixed skew shape \(\lambda/\mu\) and \(m\geq1,\) the stretching polynomial \(k\mapsto\schurS_{k\lambda/k\mu}(1^m)\) has nonnegative coefficients [JM26, Thm. 3.5]. This is the Ehrhart positivity theorem for skew Gelfand–Tsetlin polytopes, conjectured by Alexandersson and Alhajjar.
#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–Nakayama rule
Using the same notation as for the Schur polynomials, \[\schurS_{\lambda/\mu}(\xvec) = \sum_{\nu} \frac{\powerSum_{\nu}(\xvec)}{z_\nu} \sum_{T \in \BST(\lambda/\mu,\nu)} (-1)^{\mathrm{ht}(T)}\] where the sum is taken over border strip tableaux of skew shape \(\lambda/\mu\) and weight \(\nu.\)
#Hamel–Goulden identities
In [HG95], A. Hamel and I. Goulden describe a uniform framework, unifying the Jacobi–Trudi identities, the Giambelli formula and the result by A. Lascoux and P. Pragacz [LP88].
An outside decomposition of a skew diagram \(\lambda/\mu\) is a decomposition into border strips, each of which starts on the left or lower boundary of \(\lambda/\mu\) and ends on the upper or right boundary. Such a decomposition determines a common cutting strip: after translating the border strips so that cells of equal content lie on the same diagonal, each border strip appears as a segment of this cutting strip.
Given an outside decomposition \((\theta_1,\dotsc,\theta_m)\) of \(\lambda/\mu,\) let \(\theta_i\#\theta_j\) denote the segment of the cutting strip that starts on the initial diagonal of \(\theta_j\) and ends on the terminal diagonal of \(\theta_i,\) with the usual empty and zero conventions from [HG95]. Then one has \[\schurS_{\lambda/\mu}(\xvec) = \det\left[ \schurS_{\theta_i \# \theta_j}(\xvec) \right]_{1\leq i,j \leq m}.\] See [HG95] for the full notation and conventions.
Example
Let \(D=(3,3,3,1)/(1),\) and take the outside decomposition \(\Pi=(\theta_1,\theta_2)\) used in [MW09, Ex. 2.2]. The cutting strip \(\theta=\theta(\Pi)\) has contents from \(-3\) to \(2,\) with \[\theta_1=\theta[-3,2], \qquad \theta_2=\theta[-1,1].\] Thus \(p(\theta_1)=-3,\) \(q(\theta_1)=2,\) \(p(\theta_2)=-1,\) and \(q(\theta_2)=1.\) The Hamel–Goulden matrix is \[\begin{pmatrix} \schurS_{\theta[-3,2]} & \schurS_{\theta[-1,2]} \\ \schurS_{\theta[-3,1]} & \schurS_{\theta[-1,1]} \end{pmatrix},\] and hence \[\schurS_D = \det \begin{pmatrix} \schurS_{\theta[-3,2]} & \schurS_{\theta[-1,2]} \\ \schurS_{\theta[-3,1]} & \schurS_{\theta[-1,1]} \end{pmatrix}.\] Here every entry is a ribbon Schur function for a content interval in the same cutting strip \(\theta.\)
W. Y. C. Chen, G.-G. Yan, and A. L. B. Yang study the cutting strips associated with outside decompositions and prove transformation identities between the resulting Hamel–Goulden determinants [CYY05]. Their twist transformations include the Lascoux–Pragacz transformation between the Giambelli determinant and the rim-ribbon determinant as a special case.
#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: \[\sum_{\lambda} \schurS_{\lambda/\mu} = \prod_{i=1} \frac{1}{1-x_i} \prod_{1 \leq i \lt j \leq m} \frac{1}{1-x_ix_j} \sum_{\nu} \schurS_{\mu/\nu}.\]
#Various topics
#Derived Schur polynomials
For a Schur polynomial \(\schurS_\lambda,\) the derived Schur polynomials are defined by \[\schurS_\lambda(x_1+t,\dotsc,x_n+t) = \sum_i \schurS_\lambda^{(i)}(x_1,\dotsc,x_n)t^i.\] J. Ross and K.-Y. Wu conjecture a Schur-positive log-concavity property \[(\schurS_\lambda^{(i)})^2 - \schurS_\lambda^{(i-1)}\schurS_\lambda^{(i+1)}\] for these derived polynomials [RW24]. They prove the first nontrivial case for rectangles, hooks, and two further infinite families of partitions. Á. Gutiérrez and C. Krattenthaler study Schur log-concavity for sequences of symmetric functions [GK25]. They conjecture a broad class of Schur-function sequences to be Schur log-concave and prove the property for partitions with growing first part and column, relating the phenomenon to the quantum Pascal triangle.
#Gustafson–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 Chern classes, tautological bundles, and the Atiyah–Bott localization theorem, see [FRN12, Thm. 7.1].
This identity is closely related to Widom’s formula and stretched Schur polynomials; see [Ale12, Ale14].
#Cumulants of Schur polynomials
M. Dołęga has introduced the notion of cumulants of families of symmetric functions [Do19]. 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.
B. Dubrovin relates Schur polynomials to the quantization of the Hopf hierarchy and to symplectic field theory of a disk [Dub15]. In this setting the first nontrivial Hamiltonian is the cut-and-join operator, and Schur polynomials form a common eigenbasis for the commuting quantum Hamiltonians. The total SFT potential of a disk is then expanded explicitly in the Schur basis.
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