Schur Q-function

A symmetric function introduced by I. Schur [a6] in 1911 in the construction of the irreducible spin characters of the symmetric groups $S _ { n }$ (cf. Projective representations of symmetric and alternating groups). Schur $Q$-functions are analogous to the Schur functions, which play the same role for linear characters (cf. Schur functions in algebraic combinatorics). In fact, both are special cases of Hall–Littlewood functions discovered by D.E. Littlewood [a3], but see [a4] for a description of their development and subsequent generalizations, for example, Macdonald polynomials.

There are by now (as of 2000) several other definitions; the original by Schur [a6] was in terms of Pfaffians (cf. Pfaffian), a modern version of his work is [a2]. Let $\mathbf{x} = \{ x _ { 1 } , \dots , x _ { l } \}$ be a set of variables ($l \geq n$); then

\begin{equation} \tag{a1} Q ( t ) = \prod _ { i } \frac { 1 + x _ { i } t } { 1 - x _ { i } t } = \sum _ { r \geq 0 } q _ { r } t ^ { r }. \end{equation}

For $r , s \geq 0$, define

\begin{equation*} Q_{( r , s )} = q_ r q _ { s } + 2 \sum _ { i = 1 } ^ { s } ( - 1 ) ^ { i } q_{r + i} q _ { s - i}, \end{equation*}

and then $Q _ { ( s , r ) } = - Q _ { ( r , s ) }$, because $\sum _ { i = 0 } ^ { n } ( - 1 ) ^ { i } q _ { i } q _ { n - i } = 0$, as follows directly from (a1).

If $\lambda = ( \lambda _ { 1 } , \dots , \lambda _ { 2 m } )$ is a strict partition of $n$, where $\lambda _ { 1 } > \ldots > \lambda _ { 2 m } \geq 0$, then the matrix

\begin{equation*} M _ { \lambda } = ( Q _ { \langle \lambda _ { i } , \lambda _ { j } \rangle } ) \end{equation*}

is skew-symmetric, and the Schur $Q$-function $Q_\lambda$ is defined as

\begin{equation*} Q _ { \lambda } = \operatorname { Pf } ( M _ { \lambda } ), \end{equation*}

where $\text{Pf}$ stands for the Pfaffian.

An alternative purely combinatorial definition has been given by J.R. Stembridge [a7] in terms of shifted (Young) diagrams. These differ from Young diagrams (cf. Young diagram) in that there is an indentation along the diagonal. Young tableaux are replaced by marked shifted tableaux which are defined as follows. Let $P ^ { \prime }$ denote the ordered alphabet $\{ 1 ^ { \prime } < 1 < 2 ^ { \prime } < 2 < \ldots \}$; then a marked shifted tableau $T$ is a labelling of the nodes of the shifted diagram of shape $\lambda$ such that

i) the labels weakly increase along each row and down each column;

ii) each column contains at most one $k$, for each $k \geq 1$; and

iii) each row contains at most one $k ^ { \prime }$, for each $k \geq 1$. If $\gamma _ { k }$ denotes the number of nodes labelled either $k$ or $k ^ { \prime }$, then $\gamma = ( \gamma _ { 1 } , \gamma _ { 2 } , \dots )$ is the content of $T$ and if $x ^ { T } = x _ { 1 } ^ { \gamma _ { 1 } } x _ { 2 } ^ { \gamma _ { 2 } } \dots$, then

\begin{equation*} Q _ { \lambda } = \sum _ { T } x ^ { T } , \end{equation*}

summed over all marked shifted tableaux of shape $\lambda$. It is a non-trivial task to prove that this is the Schur $Q$-function. For example, if $\lambda = ( 4,2,1 )$, then the corresponding shifted diagram and a possible marked shifted tableau are

\begin{equation*} \lambda = \left. \begin{array} { l l l } { \bullet } & { \bullet } & { \bullet } & { \bullet } \\ { \square } & { \bullet } & { \bullet } & { \square } \\ { \square } & { \square } & { \bullet } & { \square } \end{array} \right. \end{equation*}

and $x ^ { T } = x _ { 1 } ^ { 3 } x _ { 2 } ^ { 4 }$.

This combinatorial definition has been a rich source of significant combinatorial results, for example, Stembridge [a7] has proved an analogue of the Littlewood–Richardson rule that describes the Schur $Q$-function expansion of $Q _ {\lambda} Q _ { \mu }$ and also gives a purely combinatorial proof for the Murnaghan–Nakayama rule for computing the irreducible spin characters of $S _ { n }$ (cf. Representation of the symmetric groups). All of this is based on a shifted version of the Robinson–Schensted–Knuth correspondence given independently by B.E. Sagan [a5] and D.R. Rowley (cf. also Robinson–Schensted correspondence).

Schur $Q$-functions also arise naturally in other contexts, for example, the characters of irreducible representations of the queer Lie super-algebra $Q ( n )$, the cohomology classes dual to Schubert cycles in isotropic Grassmannians and in polynomial solutions of the BKP-hierarchy of partial differential equations.

References

[a1]	P.N. Hoffman, J.F. Humphreys, "Projective representations of the symmetric groups" , Oxford Univ. Press (1992)
[a2]	T. Józefiak, "Characters of projective representations of symmetric groups" Exp. Math. , 7 (1989) pp. 193–247
[a3]	D.E. Littlewood, "On certain symmetric functions" Proc. London Math. Soc. , 11 : 3 (1961) pp. 485–498
[a4]	I.G. Macdonald, "Symmetric functions and Hall polynomials" , Oxford Univ. Press (1997) (Edition: Second)
[a5]	B.E. Sagan, "Shifted tableaux, Schur Q-functions and a conjecture of R. Stanley" J. Combin. Th. A , 45 (1987) pp. 62–103
[a6]	I. Schur, "Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen" J. Reine Angew. Math. , 139 (1911) pp. 155–250
[a7]	J.R. Stembridge, "Shifted tableaux and projective representations of symmetric groups" Adv. Math. , 74 (1989) pp. 87–134