#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 \(SP(\lambda/\mu).\)
For symmetry, see [Thm. 6.12, Sun86], 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, [Eq. (3.11), FK97].
#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 [Conj. 6.2, Lee19].
#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.
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 [p. 295, Lit77]; see also [p. 54, Kra98].
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.
#Schur’s \(P\) functions
Schur’s \(P\) functions were introduced by I. Schur in [Sch11] to study projective representations of the symmetric and alternating groups. Schur’s \(P\) functions \(\schurP_\lambda(\xvec)\) are indexed by strict partitions, that is, partitions with distinct parts, and are given as the specialization of the Hall–Littlewood \(P\) functions at \(t=-1.\) F. Ardila and L. G. Serrano prove Stanley’s conjecture that, for the staircase shape \(\delta_n,\) every skew Schur function \(\schurS_{\delta_n/\mu}\) expands nonnegatively in Schur’s \(P\) functions [AS12]. Their coefficients count fillings of shifted shapes; the special case \(\schurS_{\delta_n/\delta_{n-2}}\) is related to Eulerian numbers and alternating permutations. M. Fayers proves a Pieri-type rule for Schur’s \(P\) functions in [Prop. 3.6, Fay20]. For the reduction operators \(\partial_{(a)}\) and \(\partial_{(1^a)},\) one has \[\partial_{(a)}\schurP_\lambda=\partial_{(1^a)}\schurP_\lambda =\sum_\mu Hs_{\lambda/\mu}\schurP_\mu,\] where \(Hs_{\lambda/\mu}=2^N\) if \(\lambda/\mu\) is a horizontal strip and \(N\) counts columns \(c+1\) containing a box of \(\lambda/\mu\) such that column \(c\) contains none; otherwise \(Hs_{\lambda/\mu}=0.\)
#Pfaffian formula
Let \(\lambda\) be a partition with \(\ell\) distinct parts and let \(n \geq \ell.\) Then \[\schurP_\lambda(x_1,\dotsc,x_n) \coloneqq \frac{1}{(n-\ell)!} \sum_{\sigma \in \symS_n} \sigma \left( \xvec^{\lambda} \prod_{\substack{i \leq \ell \\ i \lt j \leq n } } \frac{x_i+x_j}{x_i-x_j} \right).\]
#Combinatorial formula
Given a partition \(\lambda\) with distinct parts, consider the shifted tableau of shape \(\lambda.\) We consider fillings of this diagram with entries in the alphabet \[1' \lt 1 \lt 2' \lt 2 \lt \dotsb\] such that
each row has at most one marked \(i\) for every \(i=1,2,\dotsc\)
each column has at most one unmarked \(i,\) for every \(i=1,2,\dotsc,\)
entries in rows and columns are weakly increasing and
there are no marked entries on the main diagonal.
We let \(ShSSYT(\lambda)\) denote the set of such fillings.
Example
For \(\lambda=9531\) we have the shifted diagram with an element of \(ShSSYT(\lambda).\)
Schur’s \(P\) function is defined by \[\schurP_\lambda(\xvec) \coloneqq \sum_{T \in ShSSYT(\lambda)} \xvec^T\] where \(\xvec^T\) is the product over all entries in \(T,\) and \(i\) and \(i'\) both contribute \(x_i.\)
R. C. King and A. M. Hamel study Hall–Littlewood polynomials \(P_\lambda(x;t)\) attached to arbitrary simple Lie algebras at \(t=-1\) [KH07]. In type \(A,\) the case of partitions with distinct parts recovers Schur’s \(P\) functions. For type \(C,\) they give analogous \(Q\)-functions using primed symplectic shifted tableaux, together with Weyl-group symmetry and lattice-path determinant interpretations.
An alternative combinatorial model uses semistandard decomposition tableaux, introduced in [Ser10]. See [Cho12] for additional results.
#Schur expansion
Schur’s \(P\) functions are Schur-positive; see [Wor84, Sag87, Ass18]. B. Sagan gives a proof using an insertion algorithm similar to RSK insertion that respects the Knuth relations. The proof by S. Assaf uses dual equivalence.
S.-I. Choi and J.-H. Kwon give a proof of Schur positivity using crystals; see [CK18]. A different proof using crystals is provided in [AO18].
Example (Schur expansions).
Here are the Schur expansions of \(\schurP_\lambda(\xvec)\) for various \(\lambda.\)
\( \text{Partition} \) \( \text{Schur expansion} \) \( 1 \) \( \schurS_{1} \) \( 2 \) \( \schurS_{2} + \schurS_{11} \) \( 21 \) \( \schurS_{21} \) \( 3 \) \( \schurS_{3} + \schurS_{21} + \schurS_{111} \) \( 31 \) \( \schurS_{22} + \schurS_{31} + \schurS_{211} \) \( 321 \) \( \schurS_{321} \) \( 4 \) \( \schurS_{4} + \schurS_{31} + \schurS_{211} + \schurS_{1111} \) \( 41 \) \( \schurS_{32} + \schurS_{41} + \schurS_{221} + \schurS_{311} + \schurS_{2111} \) \( 32 \) \( \schurS_{32} + \schurS_{221} + \schurS_{311} \) \( 421 \) \( \schurS_{322} + \schurS_{331} + \schurS_{421} + \schurS_{3211} \)#Littlewood–Richardson rule
The coefficients \(f_{\lambda\mu}^{\nu}\) in \[\schurP_\lambda(\xvec) \schurP_\mu(\xvec) = \sum_{\nu} f_{\lambda\mu}^{\nu} \schurP_\nu(\xvec)\] are nonnegative integers; see [Ste89]. A combinatorial description of these coefficients was first given by L. Serrano [Ser09]. Another proof is given in [Cho12]. S. Assaf gave a proof using dual equivalence [Ass18].
The coefficients \(f_{\lambda\mu}^{\nu}\) appear in the study of the type \(B\) and type \(D\) Schubert calculus of Grassmannians.
D. K. Nguyen gives a new combinatorial model for \(f_{\lambda\mu}^{\nu}\) and for the coefficients in the Schur expansion of \(\schurQ_\lambda\) [Ngu22]. Santiago Estupinan-Salamanca and O. Pechenik give an easier formula for \(f_{\lambda\mu}^{\nu}\) using a type of Yamanouchi word [EP25].
#Evacuation
For shifted tableau switching and related evacuation operations, see [CNO17].
#Schur’s \(Q\) functions
Schur’s \(Q\) functions \(\schurQ_\lambda(\xvec)\) are indexed by strict partitions and are given as the specialization of the Hall–Littlewood \(Q\) functions at \(t=-1.\) R. Stanley gives a combinatorial formula using Gelfand–Tsetlin patterns [Sta86].
Equivalently, \(\schurQ_\lambda(\xvec) = 2^{\length(\lambda)} \schurP_\lambda(\xvec).\) Thus \(\schurQ_\lambda(\xvec)\) can be realized as a sum over tableau objects by modifying the formula for Schur’s \(P\) functions and allowing diagonal entries in \(ShSSYT(\lambda)\) to be marked.
Schur’s \(Q\) functions are related to spin characters, which were part of the original motivation for studying these functions. The following is due to Schur.
Theorem (See [3.3.6, Mor62] and [(7.1), Ste89]).
Let \(\lambda\) be a partition of \(n\) with \(m\) distinct parts. The spin characters \(\zeta_{\mu}^{\lambda}\) satisfy \[\schurQ_\lambda(\xvec) = \sum_{\mu} 2^{(\length(\lambda)+ \length(\mu)+\epsilon)/2} \zeta_{\mu}^{\lambda} \frac{ \powerSum_{\mu} }{z_\mu} .\] Here, \(\mu\) runs over partitions with odd parts, and \(\epsilon\) is \(0\) or \(1\) depending on whether \(n-m\) is even or odd.
The one-part partitions have a rather explicit expression: \[\schurQ_{(n)}(\xvec) = \sum_{\lambda \vdash n} 2^{\length(\lambda)} \monomial_{\lambda}(\xvec).\]
H. Rosengren gives formulas for \(\schurQ_{\lambda}(1,q,q^2,\dotsc,q^n)\) and also shows [Prop. 3.1, Ros08] that for a partition \(\lambda\) of length \(m,\) \[\schurQ_{\lambda}(1,q,q^2,\dotsc) = \prod_{i=1}^m \frac{(-1;q)_{\lambda_i}}{ (q;q)_{\lambda_i} } \prod_{1 \leq i \lt j \leq m} \frac{q^{\lambda_j} - q^{\lambda_i}}{1-q^{\lambda_i+\lambda_j}}.\] Rosengren notes that this formula is equivalent to Kawanaka’s identity for Schur functions.
A Murnaghan–Nakayama-type formula for computing the spin characters \(\zeta_{\mu}^{\lambda}\) is given in [MO88].
A. Morris and J. Olsson give [Cor. 4.3, MO88] a formula which seems to be a spin version of the Fomin–Lulov hook formula for ribbon tableaux. This is similar to the properties of the classical irreducible character values for the symmetric group. Morris and Olsson use the abacus model to prove their results.
A Hall–Littlewood analogue of \(\schurQ_\lambda(\xvec)\) was introduced in [TZ03].
S. Cho, J. Huh, and S.-Y. Nam conjecture a classification of ribbon shapes for which the functions \(\schurQ_{\lambda/\mu}(\xvec)\) are power-sum positive [CHN20].
Y. Nishiyama studies a conjecture of H. Mizukawa, M. Nakajima, and H.-F. Yamada relating 2-reduced Schur functions and Schur \(Q\)-functions [Nis22]. The conjecture expresses certain sums involving Littlewood–Richardson coefficients and two 2-reduced Schur functions as scalar multiples of Schur \(Q\)-functions.
For more on spin characters, see [MS20].
Plethysm stability for Schur’s \(Q\)-functions is studied in [GJ24].
J. Graf and N. Jing give a Pfaffian formulation of Schur’s \(Q\)-functions which allows the indexing composition to have negative parts [GJ25]. This makes the Pfaffian construction compatible with the tableau and vertex-operator pictures and gives algebraic proofs of several standard \(Q\)-function identities. S.-J. Lee studies shifted \(t\)-Schur functions arising from the modified odd Greaves–Jing–Zhu operator [Lee26, Lee26, Lee26]. These functions are obtained from Schur’s \(Q\)-functions by the plethystic substitution \(\mathcal{Q}_\lambda(X;t)=Q_\lambda[X-tX],\) so the associated weight \(\mathcal{Q}_\lambda(X;t)P_\lambda(Y)\) is a shifted Schur weight with a virtual first alphabet. The paper gives the normalization, Pfaffian correlation kernel, Fredholm Pfaffian for the largest part, and size cumulants; at \(t=-q\) with \(q\geq 0\) the virtual alphabet becomes the positive alphabet \(X+qX.\) The companion papers establish two-row and Pfaffian Giambelli formulas, Cauchy and finite shifted-Gessel identities, and explicit transition matrices between the shifted \(t\)-Schur bases. At cyclotomic specializations those matrices have Schur-\(Q\)-positive entries, together with reciprocity, rank, and unimodality properties.
C. Defant and D. Searles introduce shifted domino functions, a type-\(B\) analogue of Schur’s \(Q\)-functions, using type-\(B\) \(0\)-Hecke modules and standard shifted domino tableaux [DS24]. These functions expand positively in T. K. Petersen’s type-\(B\) peak functions.
#Cauchy identity
B. Sagan proves [Cor. 8.3, Sag87] that \[\prod_{i,j=1}^\infty \frac{1+x_iy_j}{1-x_iy_j} = \sum_{\lambda} 2^{-\length(\lambda)} \schurQ_\lambda(\xvec)\schurQ_\lambda(\yvec),\] where the sum is over all strict partitions.
#Generalized Schur \(P\)- and \(Q\)-functions
S. Okada introduces generalized Schur \(P\)-functions and generalized Schur \(Q\)-functions associated with an admissible polynomial sequence \(F=\{f_d\}_{d\geq 0}\) [Oka19]. The functions \(P^F_\lambda(\xvec)\) are defined by a Nimmo-type Pfaffian formula, and this construction is Macdonald’s ninth variation in the \(P\)/\(Q\) setting. The ordinary Schur \(P\)- and \(Q\)-functions are recovered by taking \(f_d(u)=u^d\) and \(f_d(u)=2u^d,\) respectively. The same framework includes Ivanov’s factorial \(P\)- and \(Q\)-functions and the \(t=-1\) Hall–Littlewood functions attached to classical root systems. Okada proves Cauchy-type identities, a Józefiak–Pragacz–Nimmo formula for skew \(Q\)-functions, and a Pieri-type rule.
#Bottom Schur functions
The bottom Schur functions were introduced by P. Clifford and R. Stanley in [CS04].
Let the degree of the power-sum symmetric functions be \(1,\) so that \(\deg(\powerSum_i)=1\) and \(\deg(\powerSum_\nu) = \length(\nu).\) The Murnaghan–Nakayama rule states that \[\schurS_\lambda = \sum_{\mu} \frac{\chi^\lambda(\mu)}{z_\mu} \powerSum_\mu\] The bottom Schur function \(\hat{\schurS}_\lambda\) is defined as \[\hat{\schurS}_\lambda = \sum_{\mu : \length(\mu)=\text{lowest}} \frac{\chi^\lambda(\mu)}{z_\mu} \powerSum_\mu,\] where the sum is over partitions contributing to the lowest-degree part in the power-sum expansion.
The bottom Schur functions can be expressed as the lowest-degree term in a Jacobi–Trudi type determinant of power-sum symmetric functions. P. Clifford and R. P. Stanley prove this by deleting from the Jacobi–Trudi matrix every row and column containing a \(1,\) and then replacing each \(\completeH_i\) by \(\powerSum_i/i\) [Sec. 3, CS04]. They also prove that the span of the bottom Schur functions of degree \(n\) has a basis indexed by partitions \(\lambda \vdash n\) with \(\length(\lambda)=\mathrm{rank}(\lambda)\) [Thm. 4.3, CS04]. Consequently its dimension is the number of partitions of \(n\) into parts differing by at least \(2,\) equivalently by the Rogers–Ramanujan identity, into parts congruent to \(\pm1\) modulo \(5.\) For a related variant of \(\hat{\schurS}_\lambda,\) they show that the same coefficients occur in the power-sum expansion and in the augmented monomial symmetric-function expansion [Thm. 7.4, CS04].
Conjecture (Alexandersson, 2019).
Let \(R=(R_{\lambda\mu})\) with \(\lambda,\mu \vdash n\) be the matrix with coefficients defined by \(\powerSum_{\lambda} = \sum_\mu R_{\lambda\mu} \monomial_\mu.\)
In [CS04], the authors consider \(R-D,\) where \(D\) is the diagonal of \(R,\) and compute the dimension of the nullspace of \(R-D\) as \(n\) increases. The initial values are \[1,1,1,2,2,3,4,5,7,9,11,15,19,24,\dotsc\] They did not guess any combinatorial interpretation of these coefficients.
In 2019, an OEIS search suggested that the sequence \(a(n)\) given by A129528 matches. The sequence \(a(n)\) is defined as follows: Consider all Dyck paths, consisting of steps \((1,1)\) and \((1,-1).\) The peak-abscissa-sum of a path is the sum of the \(x\)-coordinates of all peaks. Then \(a(n)\) counts Dyck paths of any length with peak-abscissa-sum \(n.\)
#Lecture hall Schur functions
S. Corteel and J. S. Kim introduce lecture hall Schur functions in [CK18]. They prove analogues of the Jacobi–Trudi identity and its dual. Lecture hall tableaux generalize lecture hall partitions and anti-lecture hall compositions, and contain reverse semistandard Young tableaux as a limiting case. Their generating functions appear as the Schur-expansion coefficients of multivariate little \(q\)-Jacobi polynomials, and the moments of these polynomials give product formulas for lecture hall tableaux of fixed Young diagram shape. S. Corteel and J. S. Kim enumerate bounded lecture hall tableaux [CK19]. Their formulas relate these tableaux to standard and semistandard Young tableaux, and identify the count with a coefficient in a Schur expansion of \(\schurS_\lambda(m+y_1,\dotsc,m+y_n).\)
A lecture hall tableau of type \((n,\geq,\gt{})\) and shape \(\lambda/\mu\) is a filling \(T\) of the cells of \(\lambda/\mu\) with nonnegative integers such that \[\frac{T(i,j)}{n+j-i} \geq \frac{T(i,j+1)}{n+j+1-i} \qquad\text{and}\qquad \frac{T(i,j)}{n+j-i} \gt{} \frac{T(i+1,j)}{n+j-i-1}\] whenever the cells involved are present.
Example
For \(n=3\) and \(\lambda=(2,1),\) the filling
is a lecture hall tableau of type \((3,\geq,\gt{}).\) Indeed, \(2/3\geq 1/4\) along the first row, and \(2/3\gt{}1/2\) in the first column.
For asymptotics of lecture hall tableaux, see [CKN19].
#Wreath product Schur polynomials
Wreath product Schur polynomials appear in the study of \(G \wr \symS_n\); see, for example, [Mac80] and [MRW04].
For the combinatorics, see [IJS09]. These polynomials are indexed by \(r\)-tuples of partitions, \(\underline{\lambda},\) and live in the space \(\spaceSym^{\otimes r}.\) Define the wreath product Schur polynomials as \[\schurS_{\underline{\lambda}}(\xvec^{(1)},\dotsc,\xvec^{(r)}) = \prod_{i} \schurS_{\lambda^i}(\xvec^{(i)}).\] They can be realized as a sum over \(r\)-tuples of semistandard Young tableaux of shape \(\lambda^i,\) where tableau \(i\) only uses entries from the alphabet \(\xvec^{(i)}.\)
A Murnaghan–Nakayama rule is described in [MRW04]. R. Kundu and P. Ray give a combinatorial relation between certain irreducible character values of \(G\wr\symS_n\) and \(\symS_{rn}\) when \(G\) is abelian of order \(r\) [KR25]. This extends earlier character-value comparisons for the hyperoctahedral group and wreath products. J. Janopaul-Naylor and C. Ryan Vinroot study multipartition Kostka numbers \(K_{[\lambda(j)]\mu},\) defined by sequences of semistandard Young tableaux of multipartition shape and total weight \(\mu\) [JV15]. These numbers occur as multiplicities in permutation representations of wreath products, and the paper characterizes when such a multiplicity is one.
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