Symplectic and orthogonal Schur functions

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#Symplectic Schur polynomials

The even symplectic Schur polynomials were explicitly described combinatorially in [Kin76], and are characters for the Lie group $\mathrm{Sp}(2n).$ See also [Kra98]. R. Proctor gives Young-tableau and Gelfand-pattern models, together with branching rules, for characters of the classical groups [Pro94]. M. Fulmek and C. Krattenthaler give nonintersecting lattice-path proofs of Jacobi–Trudi and Giambelli-type identities for symplectic and orthogonal characters [FK97]. P. S. Campbell and A. Stokke prove hook-content formulae for symplectic and orthogonal tableaux [CS12]. These formulae give product evaluations for the corresponding tableau generating functions. S. Okada studies intermediate symplectic characters and shifted plane partitions of shifted double staircase shape [Oka20]. These characters interpolate between Schur polynomials and symplectic characters. G. Balla constructs PBW tableaux for symplectic PBW degenerate flag varieties and gives defining equations for these varieties [Bal22].

We work in the alphabet \[\xvec = (x_1,x_1^{-1},x_2,x_2^{-1},\dotsc,x_n,x_n^{-1}).\] A symplectic tableau (also known as King tableau) of shape $\lambda/\mu$ is a filling of the shape $\lambda/\mu$ with entries in \[1 \lt \overline{1} \lt 2 \lt \overline 2 \lt \dotsb \lt n \lt \overline{n}\] such that entries are weakly increasing along rows and strictly increasing along columns, and entries in row $i$ are greater than or equal to $i.$ Let $SP(\lambda/\mu)$ denote the set of such tableaux. Define the even symplectic Schur polynomials (using the definition in [Ham97]) as \[\schurSp_{\lambda/\mu}(\xvec) = \sum_{T \in SP(\lambda/\mu)} \xvec^{T},\] where entries $i$ in $T$ contribute with $x_i,$ and entries $\overline{j}$ contribute with $x_j^{-1}.$

Example

The following tableau is an element of $SP(\lambda/\mu).$

$1 $ $ 1 $ $ \overline{1} $ $ \overline{1} $ $ \overline{2} $ $ 3$ $ \overline{1} $ $ \overline{2} $ $ 3 $ $ 4$ $ 3 $ $ 3$

For symmetry, see [Sun86, Thm. 6.12], where a type of Bender–Knuth involution is defined. The proof seems to have a gap; see the discussion in [Hop20]. A complete proof is available in [Gut24].

There are also alternative tableau descriptions. For example, the Kashiwara–Nakashima tableaux carry a $U_q(\mathfrak{sp}(2m,\setC))$-crystal structure; see [AFT22].

#Bialternant formula

The even symplectic Schur functions can be expressed as \[\schurSp_\lambda(x^{\pm 1}_{1},x^{\pm 1}_{2},\dotsc, x^{\pm 1}_{n}) = \frac{ \left| x_i^{\lambda_j+n-j+1} - x_i^{-(\lambda_j+n-j+1)}\right|_{1 \leq i,j \leq n} }{ \left| x_i^{n-j+1} - x_i^{-(n-j+1)}\right|_{1 \leq i,j \leq n} }\]

The odd symplectic Schur functions have a more complicated bialternant formula; see [Oka20].

N. Jing, Z. Li, X. Pan, D. Wang, and C. Ye introduce skew odd orthogonal characters and interpolating Schur polynomials [JLPW+25]. They prove Cauchy and Jacobi–Trudi identities, describe Gelfand–Tsetlin-pattern models, and give transition formulas among symplectic, even orthogonal, and odd orthogonal characters.

#Cauchy identity

The following Cauchy identity is proved by S. Sundaram in her thesis; see [Sun86]. \[\sum_{\lambda} \schurSp_\lambda(x^{\pm 1}_{1},x^{\pm 1}_{2},\dotsc, x^{\pm 1}_{n}) \schurS_\lambda(y_1,\dotsc,y_n) = \prod_{1\leq i \lt j \leq n} (1-y_i y_j) \prod_{1\leq i \lt j \leq n} \frac{1}{(1-y_i x_j)(1-y_i x^{-1}_j)}.\]

Ronald C. King derives multiplicative generating functions for series of characters of classical Lie groups [Kin23]. The identities generalize Schur and symplectic Cauchy-type expansions to all classical types, with recurrence relations for the expansion coefficients coming from Weyl group actions.

A. Patel, H. Patel, and A. Stokke prove orthosymplectic analogues of the Cauchy and dual Cauchy identities [PPS22]. Their proofs are bijective and use insertion algorithms modeled on A. Berele’s symplectic insertion.

A. Stokke gives a Pieri rule for multiplying an orthosymplectic character by an orthosymplectic character indexed by a one-row partition [Sto18]. The resulting product rule agrees with S. Sundaram’s Pieri rule for symplectic characters.

D. Betea studies symplectic and orthogonal analogues of A. Okounkov’s Schur measure [Bet18]. He proves that these measures have determinantal correlation functions with double-contour kernels, and relates them to Toeplitz-plus-Hankel determinants and asymptotic edge kernels.

#Jacobi–Trudi

In [FK97], the following analogues of the Jacobi–Trudi identities are proved: \[\begin{aligned} \schurSp_\lambda(x^{\pm 1}_{1},x^{\pm 1}_{2},\dotsc, x^{\pm 1}_{n}) & = \frac{1}{2} \det[ \completeH_{\lambda_i - i+j}(\xvec) + \completeH_{\lambda_i - i- j + 2}(\xvec) ]_{1\leq i,j \leq \length(\lambda)} \\ & = \det[ \elementaryE_{\lambda'_j - j+i}(\xvec) - \elementaryE_{\lambda'_j - j- i}(\xvec) ]_{1\leq i,j \leq \lambda_1}. \end{aligned}\] These formulas allow us to consider $\schurSp_\lambda(\xvec)$ as a symmetric function, by instead using the infinite alphabet $x_1,x_2,\dotsc.$

Example

For example, the monomial expansions begin as follows: \[\begin{aligned} \schurSp_{4} &= \monomial_{4} + \monomial_{22} + \monomial_{31}+\monomial_{211}+\monomial_{1111} \\ \schurSp_{31} &= -\monomial_{2} - \monomial_{11} + \monomial_{22}+\monomial_{31}+2\monomial_{211}+3\monomial_{1111} \\ \schurSp_{22} &= - \monomial_{11} + \monomial_{22} + \monomial_{211}+2\monomial_{1111} \\ \schurSp_{211} &= 1 - \monomial_{2} -2 \monomial_{11} + \monomial_{211} + 3\monomial_{1111} \\ \schurSp_{1111} &= - \monomial_{11} + \monomial_{1111} \end{aligned}\] The finite symplectic alphabet must be substituted in these expressions in order for these formulas to agree with the tableau definition. For example, \[\begin{aligned} \schurSp_{11}(\xvec) &= - 1 + \monomial_{11}(\xvec)\\ \schurSp_{11}(x_1,x_1^{-1},x_2,x_2^{-1}) &= 1 + x_1^{-1}x_2^{-1} + x_1x^{-1}_2+x^{-1}_1x_2 + x_1x_2 \end{aligned}\]

#Giambelli formula

In Frobenius coordinates, \[\schurSp_{(\alpha|\beta)} = \left| \schurSp_{(\alpha_i|\beta_j)} \right|_{s \times s}\] see, for example, [FK97, Eq. (3.11)].

#Kashiwara crystals

In [Lee19], S. J. Lee defines a crystal graph structure on King tableaux, so that connected components correspond to single symplectic Schur functions. The crystal structure on $K(\mu,n),$ the set of symplectic tableaux of shape $\mu$ and maximal entry at most $n,$ is isomorphic to the crystal $B(\mu),$ the irreducible representation of $\mathfrak{sp}_{2n}$ indexed by $\mu.$

Lee also gives a crystal-preserving bijection between symplectic tableaux and oscillating tableaux.

An explicit conjecture regarding the expansion \[\schurSp_\lambda(\xvec^{\pm}) \schurS_\mu(\xvec^{\pm}) = \sum_{\nu} c^{\nu}_{\lambda\mu} \schurSp_\nu(\xvec^{\pm})\] appears in [Lee19, Conj. 6.2].

#Murnaghan–Nakayama rules

N. Kumari and A. Stokke prove Murnaghan–Nakayama rules for symplectic, orthogonal, and orthosymplectic Schur functions [KS26]. For symplectic Schur functions, the product of a symplectic power sum with $\schurSp_\mu$ is expressed as a sum of three kinds of terms: symplectic Schur functions obtained by adding a border strip to $\mu,$ those obtained by removing a border strip from $\mu,$ and an additional Weyl-denominator correction term. The orthogonal rules are analogous, while the orthosymplectic rule has a third term involving both symplectic and ordinary Schur functions.

#Root-of-unity twists

D. Littlewood’s factorization of Schur functions at variables twisted by a primitive root of unity can be expressed in terms of cores and quotients. S. P. Albion lifts these factorizations to universal characters [Alb23]. This also gives uniform proofs of related factorizations for orthogonal and symplectic characters.

N. Kumari extends character factorizations at root-of-unity twists for general linear, symplectic, orthogonal, and special orthogonal groups [Kum24]. The paper characterizes when the specialized characters vanish and factors the nonzero cases into characters of smaller classical groups, using Weyl character formulas and beta-sets of $t$-core partitions. S. P. Albion embeds the Schur, symplectic, and orthogonal factorization formulas into a family involving a parameter $z$ and a parameter $q$ [Alb25]. The proof uses $t$-cores and $t$-quotients of $z$-asymmetric partitions and relates the formulas to plethysm. A. Ayyer and N. Kumari obtain further root-of-unity factorizations for classical and universal characters [AK24], including specializations of Schur polynomials and staircase-shape factorizations.

#Further properties

There are analogues of Gelfand–Tsetlin patterns for these functions; see, for example, [Pro94, AF19, JLW24].

T. Amdeberhan, G. E. Andrews, and C. Ballantine study partition identities involving hook lengths and symplectic or orthogonal contents [AAB23]. They prove special cases of Nekrasov–Okounkov-type conjectures and relate nonzero symplectic contents to partitions into distinct even parts.

N. Jing, Z. Li, and D. Wang use vertex operators to realize the skew symplectic and skew orthogonal Schur functions of K. Koike and I. Terada [JLW24]. They derive Jacobi–Trudi identities, Gelfand–Tsetlin pattern formulas, branching rules, and Cauchy-type identities for these functions. V. Sathish Kumar and J. Torres prove a bijection between the branching models of Kwon and Sundaram using flagged hives [KT24]. Their construction uses a hive-model symmetry for Littlewood–Richardson coefficients and gives a new flagged-hive branching model.

O. Azenhas gives a combinatorial inverse to Watanabe’s type-$AII$ quantum Littlewood–Richardson map [Aze26]. Using reverse Schensted column insertion and inverse reduction of symplectic columns, this proves the $GL_{2n}(\setC)$-to-$Sp_{2n}(\setC)$ branching bijection and realizes the orthogonal-transpose symmetry on Littlewood–Richardson–Sundaram tableaux.

Z. Jin, N. Jing, Z. Li, and D. Wang introduce universal symplectic and orthogonal functions which include symplectic Schur functions, odd symplectic characters, and universal symplectic and orthogonal characters as special cases [JJLW22]. Their vertex-operator construction gives skew versions, transition formulas, and general branching rules.

N. Kumari proves a determinantal formula for orthosymplectic Schur functions analogous to the Moens–Van der Jeugt determinant for general linear Lie superalgebras [Kum24]. As a consequence, the odd symplectic characters of R. Proctor are realized as specialized orthosymplectic characters.

The even symplectic Schur functions can be expressed as a sum of skew Schur functions: \[\schurSp_\lambda(\xvec) = \sum_\alpha (-1)^{|\alpha|/2} \schurS_{\lambda/\alpha}(\xvec)\] where $\alpha$ ranges over partitions whose Frobenius coordinates are $(a_1,a_2,\dotsc, | a_1+1,a_2+1,\dotsc),$ including the empty partition.

The following branching rule is due to D. Littlewood [Lit77, p. 295]; see also [Kra98, p. 54].

Theorem (Littlewood).

\[\schurS_{\lambda}(\xvec) = \sum_{\nu} \schurSp_\nu(\xvec) \sum_{\substack{\mu\\ \mu' \text{ even}}} c^{\lambda}_{\mu,\nu}\] where the sum is over all partitions $\mu$ where the columns are even.

#Orthogonal Schur polynomials

A special orthogonal tableau of shape $\lambda/\mu$ is a filling of the shape $\lambda/\mu$ with entries in \[1 \lt \overline{1} \lt 2 \lt \overline 2 \lt \dotsb \lt n \lt \overline{n} \lt \infty\] such that (a) entries are weakly increasing along rows and all finite entries are strictly increasing along columns, (b) entries in row $i$ are greater than or equal to $i,$ and (c) there is at most one $\infty$ in every row. Let $SO(\lambda/\mu)$ denote the set of such tableaux.

Define the even orthogonal Schur polynomials using the definition in [Ham97]: \[\schurOr_{\lambda/\mu}(\xvec) = \sum_{T \in SO(\lambda/\mu)} \xvec^{T},\] where entries $i$ in $T$ contribute with $x_i$ and entries $\overline{j}$ contribute with $x_j^{-1}.$ The $\infty$-elements do not contribute.

The standard involution on symmetric functions satisfies $\omega(\schurSp_\lambda)=\schurOr_{\lambda'}.$

#Jacobi–Trudi

The orthogonal Schur polynomials satisfy the following Jacobi–Trudi identities; see [FK97, Ham97]. They follow from the symplectic identities using $\omega(\schurSp_\lambda)=\schurOr_{\lambda'}.$

\[\begin{aligned} \schurOr_\lambda(x^{\pm 1}_{1},x^{\pm 1}_{2},\dotsc, x^{\pm 1}_{n}) & =\det[ \completeH_{\lambda_i - i+j}(\xvec) - \completeH_{\lambda_i - i- j }(\xvec) ]_{1\leq i,j \leq \length(\lambda)} \\ & =\frac{1}{2}\det[ \elementaryE_{\lambda'_i - i+j}(\xvec) + \elementaryE_{\lambda'_i - i- j + 2}(\xvec) ]_{1\leq i,j \leq \lambda_1}. \end{aligned}\] Note that the functions on the right-hand side are also evaluated in the alphabet $\xvec = (x^{\pm 1}_{1},x^{\pm 1}_{2},\dotsc, x^{\pm 1}_{n}).$

More general identities, analogous to the Hamel–Goulden identities, see [Ham97].

As for the symplectic characters, one has the following expansion.

Theorem (Littlewood).

\[\schurS_{\lambda}(\xvec) = \sum_{\nu} \schurOr_\nu(\xvec) \sum_{\substack{\mu\\ \mu \text{ even}}} c^{\lambda}_{\mu,\nu}\] where the sum is over all partitions $\mu$ where the rows are even.

N. Jing, Z. Li, and D. Wang give a modern vertex-operator treatment of skew symplectic and orthogonal Schur functions [JLW24]. Their formulas include Jacobi–Trudi identities, Gelfand–Tsetlin-pattern models, branching rules, and Cauchy-type identities.

#Cauchy identity

\[\sum_{\lambda} \schurOr_\lambda(x^{\pm 1}_{1},x^{\pm 1}_{2},\dotsc, x^{\pm 1}_{n}) \schurS_\lambda(y_1,\dotsc,y_n) = \prod_{1\leq i \leq j \leq n} (1-y_i y_j) \prod_{1\leq i \lt j \leq n} \frac{1}{(1-y_i x_j)(1-y_i x^{-1}_j)}.\]

#Further properties

The orthogonal Schur functions can be expressed as a sum of skew Schur functions as \[\schurOr_\lambda(\xvec) = \sum_\beta (-1)^{|\beta|/2} \schurS_{\lambda/\beta}(\xvec)\] where $\beta$ ranges over the Frobenius coordinates $(b_1+1,b_2+1,\dotsc, | b_1,b_2,\dotsc),$ including the empty partition.

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