#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 [Fay20, Prop. 3.6]. 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 [Mor62, 3.3.6] and [Ste89, (7.1)]).
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 [Ros08, Prop. 3.1] 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 [MO88, Cor. 4.3] 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 [Sag87, Cor. 8.3] 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.
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