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A symmetric function introduced by I. Schur [[#References|[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|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|Schur functions in algebraic combinatorics]]). In fact, both are special cases of Hall–Littlewood functions discovered by D.E. Littlewood [[#References|[a3]]], but see [[#References|[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 [[#References|[a6]]] was in terms of Pfaffians (cf. [[Pfaffian|Pfaffian]]), a modern version of his work is [[#References|[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 $\lambda _ { 1 } &gt; \ldots &gt; \lambda _ { 2 m } \geq 0$, 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|Pfaffian]].

An alternative purely combinatorial definition has been given by J.R. Stembridge [[#References|[a7]]] in terms of shifted (Young) diagrams. These differ from Young diagrams (cf. [[Young diagram|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 $\{ 1 ^ { \prime } &lt; 1 &lt; 2 ^ { \prime } &lt; 2 &lt; \ldots \}$; 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 } &amp; { \bullet } &amp; { \bullet } &amp; { \bullet }  \\ { \square } &amp; { \bullet } &amp; { \bullet } &amp; { \square } \\ { \square } &amp; { \square } &amp; { \bullet } &amp; { \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 [[#References|[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|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 [[#References|[a5]]] and D.R. Rowley (cf. also [[Robinson–Schensted correspondence|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====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  P.N. Hoffman,   J.F. Humphreys,   "Projective representations of the symmetric groups" , Oxford Univ. Press  (1992) {{ZBL| 0777.20005}}</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top">  T. Józefiak,   "Characters of projective representations of symmetric groups"  ''Exp. Math.'' , '''7'''  (1989)  pp. 193–247</td></tr>
<tr><td valign="top">[a3]</td> <td valign="top">  D.E. Littlewood,   "On certain symmetric functions"  ''Proc. London Math. Soc.'' , '''11''' :  3  (1961)  pp. 485–498</td></tr>
<tr><td valign="top">[a4]</td> <td valign="top">  I.G. Macdonald,   "Symmetric functions and Hall polynomials" , Oxford Univ. Press  (1997)  (Edition: Second)</td></tr>
<tr><td valign="top">[a5]</td> <td valign="top">  B.E. Sagan,   "Shifted tableaux, Schur Q-functions and a conjecture of R. Stanley"  ''J. Combin. Th. A'' , '''45'''  (1987)  pp. 62–103</td></tr>
<tr><td valign="top">[a6]</td> <td valign="top">  I. Schur,   "Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen"  ''J. Reine Angew. Math.'' , '''139'''  (1911)  pp. 155–250</td></tr>
<tr><td valign="top">[a7]</td> <td valign="top">  J.R. Stembridge,   "Shifted tableaux and projective representations of symmetric groups"  ''Adv. Math.'' , '''74'''  (1989)  pp. 87–134</td></tr>
</table>