Symplectic and orthogonal Schur functions are treated on the classical-group Schur page. Schur \(P\)- and \(Q\)-functions and their generalizations appear on the shifted Schur page.
#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 \(\powerSum_i/i\) [CS04, Sec. 3]. They also prove that the span of the bottom Schur functions of degree \(n\) has a basis indexed by partitions \(\lambda \vdash n\) with \(\length(\lambda)=\mathrm{rank}(\lambda)\) [CS04, Thm. 4.3]. 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 [CS04, Thm. 7.4].
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.
Bibliography
- [CS04]Peter Clifford and Richard P. Stanley. Bottom Schur functions. The Electronic Journal of Combinatorics, 11(1), September 2004.
.bib
@article{CliffordStanley2004, doi = {10.37236/1820}, url2 = {https://doi.org/10.37236/1820}, year = {2004}, month = sep, publisher = {The Electronic Journal of Combinatorics}, volume = {11}, number = {1}, author = {Peter Clifford and Richard P. Stanley}, title = {Bottom {S}chur Functions}, journal = {The Electronic Journal of Combinatorics} } - [CKN19]Sylvie Corteel, David Keating and Matthew Nicoletti. Arctic curves phenomena for bounded lecture hall Tableaux. arXiv:1905.02881, 2019.
.bib
@article{CorteelKeatingNicoletti2019x, author = {Sylvie Corteel and David Keating and Matthew Nicoletti}, title = {Arctic curves phenomena for bounded lecture hall {Tableaux}}, year = {2019}, eprint = {1905.02881}, url = {https://arxiv.org/abs/1905.02881}, journal = {arXiv e-prints} } - [CK18]Sylvie Corteel and Jang Soo Kim. Lecture hall tableaux. arXiv:1804.02489, 2018.
.bib
@article{CorteelKim2018, Author = {Sylvie Corteel and Jang Soo Kim}, Title = {Lecture Hall Tableaux}, Year = {2018}, Eprint = {1804.02489}, url = {https://arxiv.org/abs/1804.02489}, journal = {arXiv e-prints} } - [CK19]Sylvie Corteel and Jang Soo Kim. Enumeration of bounded lecture hall tableaux. arXiv:1904.10602, 2019.
.bib
@article{CorteelKim2019x, author = {Sylvie Corteel and Jang Soo Kim}, title = {Enumeration of bounded lecture hall tableaux}, year = {2019}, eprint = {1904.10602}, url = {https://arxiv.org/abs/1904.10602}, journal = {arXiv e-prints} } - [IJS09]Frank Ingram, Naihuan Jing and Ernie Stitzinger. Wreath product symmetric functions. International Journal of Algebra, 3(1–4):1–19, 2009.
.bib
@article{IngramJingStitzinger2009, Author = {Frank Ingram and Naihuan Jing and Ernie Stitzinger}, Title = {Wreath product symmetric functions}, Year = {2009}, Volume = {3}, Number = {1--4}, Pages = {1--19}, url = {https://www.m-hikari.com/ija/ija-password-2009/ija-password1-4-2009/jingIJA1-4-2009.pdf}, journal = {International Journal of Algebra} } - [JV15]James Janopaul-Naylor and C. Ryan Vinroot. Kostka multiplicity one for multipartitions. arXiv:1506.07022, 2015.
.bib
@article{JanopaulNaylorVinroot2015x, author = {James Janopaul-Naylor and C. Ryan Vinroot}, title = {Kostka multiplicity one for multipartitions}, year = {2015}, eprint = {1506.07022}, url = {https://arxiv.org/abs/1506.07022}, journal = {arXiv e-prints} } - [KR25]Rijubrata Kundu and Papi Ray. A Relationship Between Character Values Of Wreath Products And The Symmetric Group. arXiv:2501.04432, 2025.
.bib
@article{KunduRay2025x, author = {Rijubrata Kundu and Papi Ray}, title = {A {R}elationship {B}etween {C}haracter {V}alues {O}f {W}reath {P}roducts {A}nd {T}he {S}ymmetric {G}roup}, year = {2025}, eprint = {2501.04432}, url = {https://arxiv.org/abs/2501.04432}, journal = {arXiv e-prints} } - [Mac80]Ian G. MacDonald. Polynomial functors and wreath products. Journal of Pure and Applied Algebra, 18(2):173–204, September 1980.
.bib
@article{Macdonald1980, doi = {10.1016/0022-4049(80)90128-0}, url2 = {https://doi.org/10.1016/0022-4049(80)90128-0}, year = {1980}, month = sep, publisher = {Elsevier {BV}}, volume = {18}, number = {2}, pages = {173--204}, author = {Ian G. MacDonald}, title = {Polynomial functors and wreath products}, journal = {Journal of Pure and Applied Algebra} } - [MRW04]Anthony Mendes, Jeffrey Remmel and Jennifer Wagner. A $\lambda$-ring Frobenius characteristic for ${G}\wr {S}_n$. The Electronic Journal of Combinatorics, 11(1), September 2004.
.bib
@article{MendesRemmelWagner2004, doi = {10.37236/1809}, url2 = {https://doi.org/10.37236/1809}, year = {2004}, month = sep, publisher = {The Electronic Journal of Combinatorics}, volume = {11}, number = {1}, author = {Anthony Mendes and Jeffrey Remmel and Jennifer Wagner}, title = {A $\lambda$-ring {F}robenius Characteristic for ${G}\wr {S}_n$}, journal = {The Electronic Journal of Combinatorics} }