SCHUR Q-FUNCTIONS AND SPIN CHARACTERS OF SYMMETRIC GROUPS I

A. O. Morris^{∗} and A. A. Abdel-Aziz^{†} Dedicated to Dominique Foata on his 60th birthday

Submitted: March 31, 1995; Accepted: November 15, 1995

1 Introduction

In a classic paper, I. Schur [6] introduced a class of symmetric functions, now called Schur Qfunctions, in order to determine the irreducible spin (projective) characters of symmetric groups. In the case of the ordinary characters of symmetric groups, going back to the early work of D. E. Littlewood and A. R. Richardson [3], the corresponding Schur functions have been used to give useful combinatorial formulae for determining explicit values for these characters (see for example [2]). Our aim in this paper is to obtain combinatorial formulae for the spin characters. We see that as always in the spin case, the formulae are considerably more complicated with a number of new interesting phenomena arising.

2 Notation and Preliminaries

2.1 Compositions and partitions

If ℓ is a positive integer, then λ = (λ_{1}, …, λ_{m}) is a composition of ℓ if λ_{i}(i = 1, …, m) are positive integers such that

∑_{i=1}^{m} λ_{i} = ℓ.

If, in addition

λ_{1} ≥ λ_{2} ≥ … λ_{m} > 0

then λ is a partition of ℓ; λ is called a strict partition if

λ_{1} > λ_{2} > … λ_{m} > 0.

When necessary, we write ℓ(λ) for m, the length of λ and |λ| for ℓ, the weight of λ. Let P (ℓ) denote the set of partitions of ℓ, SP (ℓ) the set of strict partitions of ℓ and O(ℓ) the set of partitions where all the parts λ_{i} are odd. Also, we write λ \ {λ_{i}} for the partition {λ_{1}, …, λ_{i−1}, λ_{i+1}, …, λ_{m}}, that is, with the ith part deleted with an obvious extension to the deletion of more parts.

∗Department of Mathematics, University of Wales, Aberystwyth, Dyfed SY23 3BZ

†Department of Mathematics, Faculty of Education in Abha, PO Box 9002, Abha, Saudi Arabia

Partitions are also denoted as λ = (1^{m_{1}}2^{m_{2}} …), indicating that the part i is repeated m_{i} times.

If λ ∈ P (ℓ), let

U(λ) = {μ ∈ P (|μ| ≤ ℓ)|{μ_{1}, …, μ_{ℓ(μ)}} ⊆ {λ_{1}, …, λ_{m}},

that is, U(λ) is the set of sub-partitions of λ. For r ≤ ℓ, let

U(λ, r) = {μ ∈ U(λ)|μ ∈ P (r)},

u(λ) = |U(λ)|, u(λ, r) = |U(λ, r)|. Then, clearly U(λ, ℓ) = {λ} and u(λ) ≤ 2^{ℓ(λ)} and u(λ) = 2^{ℓ(λ)} if and only if λ ∈ SP (ℓ).

Definition 2.1 A separation of λ ∈ P (ℓ) corresponding to the composition w = (w_{1}, w_{2}, …, w_{k}) of ℓ is a k-set of partitions (λ^{(1)}, λ^{(2)}, …, λ^{(k)}) such that λ^{(i)} ∈ P (w_{i}) (i = 1, …, k).

A complete set of separations (λ^{(1)}, …, λ^{(k)}) of λ corresponding to the composition w = (w_{1}, …, w_{k}) can be obtained by the following algorithm.

λ^{(1)} ∈ U(λ, w_{1}), λ^{(2)} ∈ U(λ \ λ^{(1)}, w_{2}), …, λ^{(k)} ∈ U(λ \ (λ^{(1)}, …, λ^{(k−1)}), w_{k})

where

U(λ) ⊃ U(λ \ λ^{(1)}) ⊃ U(λ \ (λ^{(1)}, λ^{(2)})) ⊃ … ⊃ U(λ \ (λ^{(1)}, …, λ^{(k)})) = ∅.

In this way, we form a tree of partitions of 0, w_{1}, …, w_{k} and corresponding to each branch of the tree is a separation of λ. For example, the separation of the partition λ = (1^{7}3^{2}5) corresponding to the composition w = (7, 5, 6) is represented by the following tree.

[figure: (1^{3}3); (3^{2}); (1^{6}); (15); (15); (1^{3}3); (3^{2}); (1^{2}3); (1^{5}); (5); (1^{5}); (1^{2}3); (5); (5); (1^{2}5); (13^{2}); (1^{4}3); (1^{7}); ∅; 6; 5; 7]

2.2 Hall-Littlewood Symmetric Functions

Let x = {x_{1}, x_{2}, …, x_{ℓ}} be a set of ℓ independent indeterminates and t an independent parameter. If α = (α_{1}, α_{2}, …, α_{ℓ}) ∈ Z^{ℓ}, write e^{α} := x_{1}^{α_{1}} … x_{ℓ}^{α_{ℓ}} and

θ_{ℓ}(x, t) = ∏_{1≤i<j≤ℓ} \frac{x_{i} − tx_{j}}{x_{i} − x_{j}} .

If λ is a partition with ℓ(λ) = m ≤ ℓ, put λ_{i} = 0 for i = m + 1, …, ℓ. Then the Hall-Littlewood symmetric functions P_{λ}(x; t) are defined by

P_{λ}(x; t) = ν_{λ}(t)^{−1} ∑_{w∈S_{ℓ}} w(e^{λ}θ_{ℓ}(x; t)),  (2.1)

where S_{ℓ} is the symmetric group acting on x,

ν_{λ}(t) = ∏_{i≥0} ν_{n_{i}}(t), ν_{r}(t) = (1 − t)^{−r} ∏_{i=1}^{r}(1 − t^{j})

and n_{i} := n_{i}(λ) is the multiplicity of i in λ. Alternatively, as shown by Macdonald [4]

P_{λ}(x; t) = ∑_{w∈S_{ℓ}/S_{ℓ}^{λ}} w(e^{λ}θ_{ℓ}(x; t)),  (2.2)

where S_{ℓ}^{λ} is the subgroup of w ∈ S_{ℓ} such that λ_{w(i)} = λ_{i} (1 ≤ i ≤ ℓ), and the summation is taken over left coset representatives of S_{ℓ}^{λ} in S_{ℓ}. In fact, S_{ℓ}^{λ} = ∏_{i≥0} S_{n_{i}}. We are going to be more concerned with another version of the Hall-Littlewood symmetric functions, namely

Q_{λ}(x; t) := b_{λ}(t)P_{λ}(x; t),  (2.3)

where b_{λ}(t) := ∏_{i≥1} ∏_{j=1}^{n_{i}}(1 − t^{j}). In particular, we put q_{k}(x; t) = Q_{(k)}(x; t) and note that from (2.1) we have

q_{k}(x; t) = (1 − t) ∑_{i=1}^{ℓ} x_{i}^{k}β_{i}(x; t),  (2.4)

where

β_{i}(x; t) = ∏_{{j=1 \atop i≠j}} \frac{x_{i} − tx_{j}}{x_{i} − x_{j}} (i = 1, 2, …, ℓ).

Macdonald [4] has given the following useful recurrence relation for the Q_{λ}(x; t).

Q_{λ}(x; t) = (1 − t) ∑_{i=1}^{ℓ} x_{i}^{λ_{1}}β_{i}(x; t)Q_{λ\{λ_{1}}}(x \ {x_{i}}; t).  (2.5)

We are, in particular, interested in the case corresponding to t = −1, as these are the so-called Schur Q-functions which were introduced by I. Schur [6] to deal with the spin characters of symmetric groups. In this case, we have

\begin{matrix}(i) If λ ∈ P (ℓ), then b_{λ}(−1) = \{ \begin{matrix}2^{ℓ(λ)} & ifλ ∈ SP (ℓ) , \\ 0 & otherwise, \end{matrix} \\ (ii) Q_{λ}(x; −1) = \{ \begin{matrix}2^{ℓ(λ)}P_{λ}(x; −1) if λ ∈ SP (ℓ) , \\ 0 otherwise, \end{matrix} \\ (iii) q_{k}(x; −1) = 2 ∑_{i=1}^{ℓ} x_{i}^{k}β_{i}(x; −1) (k = 1, 2, …). \end{matrix} \}  (2.6)

From now on, unless otherwise stated, we assume that λ ∈ SP (ℓ). Furthermore, in order to simplify the notation, we write Q_{λ}(x) for Q_{λ}(x, −1), q_{k}(x) for q_{k}(x, −1), β_{k}(x) for β_{k}(x, −1), θ_{ℓ}(x) for θ_{ℓ}(x, −1), etc.

I. Schur [6] defines Q-functions by a recursive formula and then determines a closed formula similar to (2.2) above. We reverse the process and show that the Q_{λ}(x) as defined above satisfy certain properties, including the recurrence properties used in I. Schur’s definition.

If λ = (λ_{1}, …, λ_{m}) ∈ SP (ℓ), then from (2.2) and (2.6)(ii), we have

Q_{λ}(x) = 2^{m} ∑_{w∈S_{ℓ}/S_{ℓ−m}} w(e^{λ}θ_{ℓ}(x)),  (2.7)

where S_{ℓ−m} is the subgroup of S_{ℓ} which fixes x_{m+1}, …, x_{ℓ} and the summation is over all left coset representatives of S_{ℓ−m} in S_{ℓ}. Thus, we have

Q_{λ}(x) = 2^{m} ∑_{w∈S_{ℓ}/S_{ℓ−m}}(−1)^{\frac{m(m−1)}{2}} w (e^{λ} \frac{θ_{m}(x)}{∏_{i=1}^{m} β_{i}(x)}) ,  (2.8)

which is the form of the Q-function obtained by I. Schur.

We now proceed to show that the recurrence relations used to define Schur’s Q-functions are satisfied. In order to do this, we need the following which was indeed proved by I. Schur [6, p.226].

θ_{m}(x) = ∑_{i=2}^{m}(−1)^{i}θ_{2}(x_{1}, x_{i})θ_{m−2}(x \ {x_{1}, x_{i}}) if m is even  (2.9)

and

θ_{m}(x) = ∑_{i=1}^{m}(−1)^{i−1}θ_{m−1}(x \ {x_{i}}) if m is odd.  (2.10)

Theorem 2.2 Let λ = (λ_{1}, …, λ_{m}) ∈ SP (ℓ). Then

(i) Q_{(r,s)}(x) = q_{r}(x)q_{s}(x) + 2 ∑_{i=1}^{s}(−1)^{s}q_{r+i}(x)q_{s−i}(x).

(ii) If m is even

Q_{λ}(x) = ∑_{i=2}^{m}(−1)^{i}Q_{(λ_{1},λ_{i})}(x)Q_{λ\{λ_{1},λ_{i}}}, (x).

(iii) If m is odd

Q_{λ}(x) = ∑_{i=1}^{m}(−1)^{i−1}q_{λ_{i}}(x)Q_{_{λ}\{λ_{i}}}(x).

Proof (i) From (2.5) we have

Q_{(r,s)}(x) = ∑_{i=1}^{ℓ} x_{i}^{r}β_{i}(x)q_{s}(x \ {x_{i}})
= ∑_{i=1}^{ℓ} x_{i}^{r}β_{i}(x) (∑_{j=1}^{ℓ} x_{j}^{s}β_{j}(x)\frac{x_{j}−x_{i}}{x_{j}+x_{i}} )
= ∑_{i=1}^{ℓ} x_{i}^{r}β_{i}(x) (∑_{j=1}^{ℓ} x_{j}^{s}β_{j}(x) (1 − \frac{2x_{i}}{x_{j}+x_{i}} )) ,

from which we deduce that

Q_{(r,s)}(x) = q_{r}(x)q_{s}(x) − 2 ∑_{i=1}^{ℓ} x_{i}^{r+1}β_{i}(x) (∑_{j=1}^{ℓ} x_{j}^{s}β_{j}(x)\frac{1}{x_{j} + x_{i}} ) .  (2.11)

A similar calculation, but now writing \frac{x_{j}−x_{i}}{x_{j}+x_{i}} = −1 + \frac{2x_{j}}{x_{j}+x_{i}} gives

Q_{(r+1,s−1)}(x) = −q_{r+1}(x)q_{s−1}(x) + 2 ∑_{i=1}^{ℓ} x_{i}^{r−1}β_{i}(x) (∑_{j=1}^{ℓ} x_{j}^{s}β_{j}(x)\frac{1}{x_{j} + x_{i}} ) .  (2.12)

Now, adding (2.11) and (2.12), we have

Q_{(r+1,s−1)}(x) = q_{r}(x)q_{s}(x) − q_{r+1}(x)q_{s−1}(x) − Q_{(r,s)}(x).

A similar calculation to the above shows that

Q_{(r,1)}(x) = q_{r}(x)q_{1}(x) − 2q_{r+1}(x)

and the proof is now completed by induction on s. (ii) If m is even, by substituting (2.9) in (2.8), we obtain

Q_{λ}(x) = 2^{m} ∑_{w∈S_{ℓ}/S_{ℓ−m}}(−1)^{m(m−1)/2}w (\frac{e^{λ} ∑_{i=2}^{m}(−1)^{i}θ_{2}(x_{1},x_{i})θ_{m−2}(x\\{x_{1},x_{i}\})}{∏_{j=1}^{m} β_{j}(x)})
= ∑_{i=2}^{m}(−1)^{i} (2^{2} ∑_{w∈S_{ℓ}/S_{ℓ−2}} −w (e^{(λ_{1},λ_{i})} \frac{θ_{2}(x_{1},x_{i})}{β_{1}(x)β_{i}(x)}))

(2^{m−2} ∑_{w∈S_{ℓ}/S_{ℓ−m+2}}(−1)^{(m−2)(m−3)/2}w (_{\frac{(e^{(λ\\{λ_{1},λ_{i}\})}θ_{m−2}(x\\{x_{1},x_{i}\})}{∏m ; {j=2 \atop j≠i} β_{j}(x)}}))

= ∑_{i=2}^{m}(−1)^{i}Q_{(λ_{1},λ_{i})}(x)Q_{λ\{λ_{1},λ_{i}}}(x)

The proof of (iii) is similar, but using (2.10). □

Macdonald [4, p.109] has shown that the inductive formula (2.5) can be used to extend the definition of the Schur Q-functions to any Q_{μ}(x), where μ = (μ_{1}, …, μ_{k}) ∈ Z^{k}, for any k. Furthermore, Theorem 2.1(i) has been used to reduce such a Q_{μ}(x) to a linear combination of Q_{λ}(x) such that λ ∈ SP (ℓ) (see also Morris [5]). The reduction rules are as follows.

Theorem 2.3 (i) Q_{(...,r,s,...)}(x) = − − Q_{(...,s,r,...)}(x) with Q_{(...,i,...,i,...)}(x) = 0. (ii) Q_{...,r,−r,...)}(x) = 0, Q_{(μ_{1},...,μ_{i−1},−r,r,μ_{i+1},...,μ_{n})}(x) = 2(−1)^{r}Q_{(μ_{1},...,μ_{i−1},μ_{i+1},...,μ_{n})}(x); in particular Q_{(s,r)}(x) = −Q_{(r,s)}(x), r ≠ s and Q_{(−r,r)}(x) = 2(−1)^{r}.

A consequence of (i) is that if λ = (λ_{1}, …, λ_{m}) ∈ SP (ℓ), then

Q_{λ}(x) = ϵ(σ)Q_{(λ_{σ(1)},...,λ_{σ(m)})}  (2.13)

for all σ ∈ S_{m}, where ϵ(σ) is the sign of the permutation σ.

3 Further Results on Schur Q-functions

¿From now on, it will be essential that it is clear on what sets our symmetric groups act, thus the notation will be modified. We denote by S\underline{m} the symmetric group which acts on the set \underline{m} = \{1, 2, …, m\} and S_{\underline{m}\\underline{k}} the symmetric group which acts on \underline{m} \ \underline{k} = \{k + 1, …, m\}. Also, we need to determine certain explicit left coset representations of certain groups. These are given by the following.

Lemma 3.1 For k = 2, …, m, put w_{k} = (2k) ∈ S\underline{m} and for k = 1, 2, …, m, put u_{k} = (1k) ∈ S\underline{m}. Then

(i) \{w_{k}|k = 2, …, m\} is a set of left coset representatives of S_{\underline{m}\\underline{2}} in S_{\underline{m}\\underline{1}}.

(ii) \{u_{k}|k = 1, …, m\} is a set of left coset representatives of S_{\underline{m}\\underline{1}} in S\underline{m}.

We can now use the following lemma to give an alternative form for the Q_{λ}(x).

Lemma 3.2 If λ = (λ_{1}, …, λ_{m}) ∈ SP (ℓ), then

Q_{λ}(x) = ∑_{η∈S_{\underline{m}\\underline{1}}/S_{\underline{m}\\underline{2}}} ϵ(η)Q_{(λ_{1},λ_{η(2)})}(x)Q_{(λ_{η(3)},...,λ_{η(m)})}(x)  (3.14)

if m is even, and

Q_{λ}(x) = ∑_{η∈S\underline{m}/S_{\underline{m}\\underline{1}}} ϵ(η)q_{η(1)}(x)Q_{(λ_{η(2)},...,λ_{η(m)})}(x)  (3.15)

if m is odd.

Proof If η ∈ S_{\underline{m}\\underline{1}}, then by Lemma 3.1(i) we have η = w_{k}ν_{k}, where w_{k} = (2k), ν_{k} ∈ S_{\underline{m}\\underline{2}} for some 2 ≤ k ≤ m, and so η =

(2k) ( \begin{matrix}3 & … & k & … & s & … & m \\ η(3) & … & η(k) & … & η(2) & … & η(m) \end{matrix} ) = ( \begin{matrix}2 & 3 & … & k & … & s & … & m \\ k & η(3) & … & η(k) & … & 2 & … & η(m) \end{matrix} )

= (k, η(k)) ( \begin{matrix}2 & 3 & … & k & … & s & … & m \\ η(k) & η(3) & … & η(2) & … & η(s) & … & η(m) \end{matrix} ) ,

the electronic journal of combinatorics 3 (2) (1996), #R20 7

where η(2) = k and η(s) = 2. Hence, if m is even

∑_{η∈S_{\underline{m}\\underline{1}}/S_{\underline{m}\\underline{2}}} ϵ(η)Q_{(λ_{1},λ_{η(n)})}(x)Q_{(λ_{η(3)},...,λ_{η(m)})}(x)

= ∑_{k=2}^{m} ϵ(k, η(k))ϵ ( \begin{matrix}2 & 3 & … & k & … & s & … & m \\ η(k) & η(3) & … & η(2) & … & η(s) & … & η(m) \end{matrix} ) Q_{(λ_{1},λ_{k})}(x)Q_{(λ_{η(3)},...,λ_{η(m)})}(x)

= ∑_{k=2}^{m} ϵ(k, η(k))Q_{(λ_{1},λ_{k})}(x)Q_{(λ_{3},...,λ_{2},...,λ_{m})}(x),

with λ_{2} in the k position in Q_{(λ_{3},...,λ_{2},...,λ_{m})}(x), where we have used (2.13). Now, by using Theorem 2.3(i) repeatedly, we have that the right hand side of the above reduces to

∑_{k=2}^{m}(−1)^{k−2}Q_{(λ_{1},λ_{k})}(x)Q_{λ\{λ_{1},λ_{k}}}(x) = Q_{λ}(x)

by Theorem 2.2(ii). A similar argument can be used to prove (ii). □ Lemma 3.3 If m is even and λ ∈ SP (ℓ), then

Q_{λ}(x) = \frac{1}{m} ∑_{σ∈S\underline{m}/S_{\underline{m}\\underline{2}}} ϵ(σ)Q_{(λ_{σ(1)},λ_{σ(2)})}(x)Q_{(λ_{σ(3)},...,λ_{σ(m)})}(x)

Proof If m is even, then from (2.13), Theorem 2.2(ii) and Lemma 3.2, we have

Q_{λ}(x) = \frac{1}{m} ∑_{σ∈S\underline{m}/S_{\underline{m}\\underline{1}}} ϵ(σ)Q_{(λ_{σ(1)},...,λ_{σ(m)})}(x)
= \frac{1}{m} ∑_{σ∈S\underline{m}/S_{\underline{m}\\underline{1}}} ϵ(σ) ∑_{i=2}^{m}(−1)^{i}Q_{(λ_{σ(1)},λ_{σ(i)})}(x)Q_{(λ_{σ(2)},...,λ_{σ(m)})\\{λ_{σ(1)},λ_{σ(i)}\}}(x)
= \frac{1}{m} ∑_{σ∈S\underline{m}/S_{\underline{m}\\underline{1}}} ϵ(σ) ∑_{η∈S_{\underline{m}\\underline{1}}/S_{\underline{m}\\underline{2}}} ϵ(η)Q_{(λ_{ση(1)},λ_{ση(2)})}(x)Q_{(λ_{ση(3)},...,λ_{ση(m)})}(x)
= \frac{1}{m} ∑_{σ∈S\underline{m}/S_{\underline{m}\\underline{2}}} ϵ(σ)Q_{(λ_{σ(1)},λ_{σ(2)})}(x)Q_{(λ_{σ(3)},...,λ_{σ(n)})}(x).
□

We can now use this result as the basis for an inductive proof of the following. Theorem 3.4 (i) If m = 2μ is even, then

Q_{λ}(x) = \frac{1}{2^{μ}μ!} ∑_{σ∈S\underline{m}} ϵ(σ)Q_{(λ_{σ(1)},λ_{σ(2)})}(x)Q_{(λ_{σ(3)},λ_{σ(4)})}(x) … Q_{(λ_{σ(m−1)},λ_{σ(m)})}(x)

(ii) If m = 2μ + 1 is odd, then

Q_{λ}(x) = \frac{1}{2^{μ}μ!} ∑_{σ∈S\underline{m}} ϵ(σ)q_{λ_{σ(1)}}(x)Q_{(λ_{σ(2)},λ_{σ(3)})}(x)Q_{(λ_{σ(4)},λ_{σ(5)})}(x) … Q_{(λ_{σ(m−1)},λ_{σ(m)})}(x)

Proof (i) The proof is by induction on m. If m = 1, then clearly

Q_{(λ_{1},λ_{2})}(x) = \frac{1}{2} ∑_{σ∈S_{2}} ϵ(σ)Q_{(λ_{σ(1)},λ_{σ(2)})}(x).

If m > 1, then by Lemma 3.3 and the inductive assumption, we have

Q_{λ}(x) = \frac{1}{2μ} ∑_{σ∈S\underline{m}/S_{\underline{m}\\underline{2}}} ϵ(σ)Q_{(λ_{σ(1)},λ_{σ(2)})}(x)\frac{1}{2^{μ−1}(μ−1)!} ∑_{ρ∈S_{(\underline{m})\σ(2)}} ϵ(ρ)Q_{(λ_{ρσ(3)},λ_{ρσ(4)})}(x)
… Q_{(λ_{ρσ(m−1)},λ_{ρσ(m)}}(x)
= \frac{1}{2^{μ}μ!} ∑_{σ∈S\underline{m}/S_{\underline{m}\\underline{2}}} ϵ(σ)Q_{(λ_{σ(1)},λ_{σ(2)})}(x) ∑_{σ′∈S_{\underline{m}\2}} ϵ(σ′)Q_{λ_{σ′(3)},λ_{σ′(4)})}(x)
… Q_{λ_{(σ′(m−1)},λ_{σ′(m)})}(x)
= \frac{1}{2^{μ}μ!} ∑_{σ∈S_{m}} ϵ(σ)Q_{(λ_{σ(1)},λ_{σ(2)})}(x)Q_{(λ_{σ(3)},λ_{σ(4)})}(x) … Q_{(λ_{σ(m−1)},λ_{σ(m)})}(x),

as required. The proof of (ii) is similar. □

The above result can be interpreted in terms of wreath products of certain groups; in fact, the hyperoctahedral group.

If λ = (λ_{1}, …, λ_{m}) where m = 2μ is even, let

M = {{λ_{1}, λ_{2}}, {λ_{3}, λ_{4}}, …, {λ_{m−1}, λ_{m}}}

and if σ ∈ S_{m}, let

σ(M) = {{λ_{σ(1)}, λ_{σ(2)}}, {λ_{σ(3)}, λ_{σ(4)}}, …, {λ_{σ(m−1)}, λ_{σ(m)}}}.

If H = {σ ∈ S_{m}|σ(M) = M}, then H is a subgroup of S_{m} of order 2^{μ}μ!. In fact, if π = (12)(34) … (m − 1, m) ∈ S_{m}, then H is the centralizer C_{S_{m}}(π) of π in S_{m}, which implies that H ≅ S_{2} ≀S_{m} (see [2,p.135]), the wreath product of S_{2} by S_{m}, which is the hyperoctahedral group or the Weyl group of type B_{μ}. Thus, the above results can be rewritten as

Theorem 3.5 For m ≥ 1, we have (i) if m = 2μ is even

Q_{λ}(x) = ∑_{σ∈S_{m}\S_{2}≀S_{μ}} ϵ(σ)Q_{(λ_{σ(1)},λ_{σ(2)})}(x) … Q_{(λ_{σ(m−1)},λ_{σ(m)})}(x)

(ii) if m = 2μ + 1 is odd

Q_{λ}(x) = ∑_{σ∈S_{m}\S_{2}≀S_{μ}} ϵ(σ)q_{λ_{σ(1)}}Q_{(λ_{σ(2)},λ_{σ(3)})}(x) … Q_{(λ_{σ(m−1)},λ_{σ(m)})}(x)

4 Application to the Spin Characters of Symmetric Groups

As was mentioned earlier, Schur Q-functions were first introduced by Schur [6] in his work on the spin (projective) characters of symmetric groups (see also [1]). He showed that the irreducible spin characters of S_{ℓ} were parameterised by strict partition λ ∈ SP (ℓ), but that this correspondence is not one-to-one. In fact, if we say that λ is even (odd) if ℓ − ℓ(λ) is even (odd), there is one irreducible self-associate spin character ζ^{λ} corresponding to λ ∈ SP (n) if λ is even and two irreducible associate spin characters ζ^{λ} and ζ^{λ′} if λ is odd. The connection with Schur Q-functions is given by

Q_{λ}(x) = ∑_{π∈O(ℓ)} 2^{\frac{1}{2} (ℓ(λ)+ℓ(π)+ϵ(λ))}z_{π}^{−1}ζ_{π}^{λ}p_{π}(x)  (4.16)

where z_{π} = ∏_{i≥1} i^{m_{i}}m_{i}!, where m_{i} is the multiplicity of i in the partition π = (1^{m_{1}}3^{m_{3}} …), p_{π}(x) = p_{1}(x)^{m_{1}}p_{3}(x)^{m_{3}} … (p_{i}(x) = ∑_{k=1}^{ℓ} x_{k}^{i}), ζ_{π}^{λ} is the value of the character ζ^{λ} at the class π and

ϵ(λ) = \{ \begin{matrix}0 & if ℓ − ℓ(λ) is even \\ 1 & if ℓ − ℓ(λ) is odd. \end{matrix}

This formula gives the value of ζ^{λ} on the classes π ∈ O(ℓ) only; however, if λ is even ζ_{π}^{λ} = 0 if π ∈ P (ℓ) \ O(ℓ) and if λ is odd ζ_{π}^{λ} = ζ_{π}^{λ′} = ζ_{π}^{λ′} if π ∈ O(ℓ), while if π ∉ O(ℓ), ζ_{π}^{λ} ≠ 0 only if π = λ and then ζ_{λ}^{λ} is given explicitly by

ζ_{λ}^{λ} = (−1)^{(−ℓ(λ)+1)}\sqrt{\frac{λ_{1}λ_{2} … λ_{ℓ(λ)}}{2}}  (4.17)

and ζ_{λ}^{λ′} = −ζ_{λ}^{λ}. Thus, formula (4.16) gives all the information required. In addition, we require

ζ_{π}^{(ℓ)} = 2^{\frac{1}{2} (ℓ(σ)−1−ϵ)}  (4.18)

where

ϵ = \{ \begin{matrix}1 & if ℓ is odd \\ 0 & if ℓ is even \end{matrix}

and

q_{ℓ}(x) = ∑_{π∈O(ℓ)} 2^{ℓ(π)}z_{π}^{−1}p_{π}(x).  (4.19)

Both results are originally due to Schur [6]; in fact (4.19) is (4.16) for the particular case λ = (ℓ) using the value given by (4.18) for the character ζ_{π}^{(ℓ)}.

Our intention now is to give a recursive formula for the calculation of the ζ_{π}^{λ} based on the preceding work on Schur Q-functions.

We shall assume from now on that π = (1^{m_{1}}3^{m_{3}} …) ∈ O(ℓ). Let {π^{1}, π^{2}, …, π^{m}} be a separation of π; then we note that

\frac{z_{π}}{z_{π^{1}}z_{π^{2}} … z_{π^{m}}} = \frac{y_{π}}{y_{π^{1}}y_{π^{2}} … y_{π^{m}}} ,  (4.20)

where y_{π} = ∏_{i≥1} m_{i}! (recall that z_{π} = ∏_{i≥1} i^{m_{i}}m_{i}!).

We first give an explicit formula for ζ_{π}(r, s) (r > s), that is for two part partitions. We prove Theorem 4.1 If λ = (r, s) ∈ SP (ℓ) (r > s), π ∈ O(ℓ), then

ζ_{π}^{(r,s)} = 2^{\frac{1}{2} (ℓ(π)−2−ϵ)}y_{π} \{ ∑_{(π_{r},π_{s})=π} y_{π_{r}}^{−1}y_{π_{s}}^{−1} + 2 ∑_{j=1}^{s}(−1)^{j} ( ∑_{(π_{r+j},π_{s−j})} y_{r+j}^{−1}y_{s−j}^{−1})\} ,

where the sums are taken over all separations (π_{r+j}, π_{s−j}) of π corresponding to the composition (partition) (r + j, s − j) and ϵ = 0(1) if ℓ is even (odd).

Proof For the partition (r, s) ∈ SP (ℓ), (4.16) becomes

Q_{(r,s)}(x) = ∑_{π∈O(ℓ)} 2^{\frac{1}{2} (ℓ(π)+2+ϵ)}z_{π}^{−1}ζ_{π}^{(r,s)}p_{π}(x),  (4.21)

where ϵ = 0(1) if ℓ is even (odd). Now, by substituting (4.19) and (4.21) in Theorem 2.2(i), we obtain

∑_{π∈O(ℓ)} 2^{\frac{1}{2} (ℓ(π)+2+ϵ)}z_{π}^{−1}ζ_{π}^{(r,s)}p_{π}(x) = ( ∑_{π_{r}∈O(r)} z_{π_{r}}^{−1}2^{ℓ(π_{r})}p_{π_{r}}(x)) ( ∑_{π_{s}∈O(s)} z_{π_{s}}^{−1}2^{ℓ(π_{s})}p_{π_{s}(x)})

+2 ∑_{j=1}^{s}(−1)^{j} ( ∑_{π_{r+j}∈O(r+j)} z_{π_{r+j}}^{−1}2^{ℓ(π_{r+j})}p_{π_{r+j}}(x))

= ∑_{(π_{r},π_{s})=π∈O(ℓ)} z_{π_{r}}^{−1}z_{π_{s}}^{−1}2^{ℓ(π)}p_{π}(x) + 2 ∑_{j=1}^{s}(−1)^{j} ∑_{(π_{r+j},π_{s−j})=π∈O(ℓ)} z_{π_{r+j}}^{−1}z_{π_{s−j}}^{−1}2^{ℓ(π)}p_{π}(x),

where the sums are taken over all separations (π_{r+j}, π_{s−j}) of π corresponding to the composition (r + j, s − j). Now, by comparing the coefficients of p_{π}(x) on both sides of this equation and using (4.19), we obtain the required result. □

Thus, the calculation of ζ_{π}^{(r,s)} for π ∈ O(ℓ), is reduced to a calculation involving the separation of partitions and is given in combinatorial terms. Example: We calculate ζ_{(1^{6}3)}^{(6,3)}. The separations of (1^{6}3) corresponding to the composition (6, 3), (7, 2), (8, 1) and (9) are ({(1^{6}), (3)}, {(1^{3}3), (1^{3})}), {(1^{4}3), (1^{2})}, {(1^{5}3), (1)} and {(1^{6}3), ∅} respectively. Thus

\begin{aligned}χ_{(1^{6}3)}^{(6,3)} & = 2^{2}6! \{(\frac{1}{6!} + \frac{1}{3!3!}) + 2 (−\frac{1}{4!2!} + \frac{1}{5!} − \frac{1}{6!})\} \\ & = 4. \end{aligned}

We now use the expansion of the Schur Q-functions Q_{λ}(x) in terms of products of Schur Qfunctions corresponding to two-part partitions to give a combinatorial formula for ζ_{π}^{λ} in the general case. We use Theorem 3.5 (or Theorem 3.4). We prove

Theorem 4.2 (i) If m = 2μ is even and λ = (λ_{1}, …, λ_{m}) ∈ SP (ℓ), π = (1^{m_{1}}3^{m_{3}} …) ∈ O(ℓ), I = {1, 3, 5, …, m − 1}, then

ζ_{π}^{λ} = ∑_{σ∈S_{m}/S_{2}≀S_{μ}} ϵ(σ) \{ ∑_{(π_{1}^{σ},π_{3}^{σ},...)=π} y_{π} (∏_{i∈I} y_{π_{i}^{σ}}^{−1}) 2^{\frac{1}{2} (−ϵ+∑_{i∈I} ϵ_{i}^{σ})} ∏_{i∈I} ζ_{π_{i}^{σ}}^{(λ_{σ(i)},λ_{σ(i+1)})}\} ,

where for each i ∈ I, π_{i}^{σ} = π_{(λ_{σ(i)},λ_{σ(i+1)})} ∈ O(|λ_{σ(i)}| + |λ_{σ(i+1)}|), ϵ_{i}^{σ} = 0(1) if |λ_{σ(i)}| + |λ_{σ(i+1)}| is even (odd), ϵ = 0(1) if ℓ is even (odd), and the sum is taken over all separations (π_{1}^{σ}, π_{3}^{σ}, …, π_{m−1}^{σ}) of π. (ii) If m = 2μ + 1 is odd and λ = (λ_{1}, …, λ_{m}) ∈ SP (ℓ), π = (1^{m_{1}}3^{m_{3}} …) ∈ O(ℓ), J = {2, 4, …, m − 1}, then

ζ_{π}^{λ} = ∑_{σ∈S_{m}/S_{2}≀S_{μ}} ϵ(σ) {  ∑_{(π_{λ_{σ(1)}},π_{2}^{σ},π_{4}^{σ},...,π_{m−1}^{σ})=π} y_{π}y_{π_{λ_{σ(1)}}}^{−1} ∏_{j∈J} y_{π_{j}^{σ}}^{−1}

2^{\frac{1}{2} (−ϵ+ϵ_{σ(1)}+∑_{j∈J} ϵ_{j}^{σ}))}ζ_{π_{λ_{σ(1)}}}^{(λ_{σ(1)})} ∏_{j∈J} ζ_{π_{j}^{σ}}^{(λ_{σ(j)},λ_{σ(j+1)}}),

where for each j ∈ J, π_{j}^{σ} = π_{(λ_{σ(j)},λ_{σ(j+1)})} ∈ O(|λ_{σ(j)}| + |λ_{σ(j+1)}|), ϵ_{j}^{σ} = 0(1) if |λ_{σ(j)}| + |λ_{σ(j+1)}| is even (odd), ϵ_{σ(1)} = 0(1) if |λ_{σ(1)}| is odd (even), ϵ = 0(1) if ℓ is odd (even) and the summation is over all separations (π_{λ_{σ(1)}}, π_{2}^{σ}, π_{4}^{σ}, …, π_{m−1}^{σ}) of π corresponding to the composition.

Proof We prove (i) only; the proof of (ii) is similar. ¿From Theorem 3.5(i) and (4.16) we obtain

∑_{π∈O(ℓ)} 2^{\frac{1}{2} (m+ℓ(π)+ϵ)}z_{π}^{−1}ζ_{π}^{λ}p_{π}(x)

= ∑_{σ∈S_{m}/S_{2}≀S_{μ}} ϵ(μ) ∏_{i∈I} ∑_{π_{i}^{σ}} 2^{\frac{1}{2} (2+ℓ(π_{i}^{σ})+ϵ_{i}^{σ})}z_{π_{i}^{σ}}^{−1}ζ_{π_{i}^{σ}}^{(λ_{σ(i)},λ_{σ(i+1)}}p_{π_{i}^{σ}}(x).

Since, if {π^{1}, π^{2}, π^{3}, …, π^{m}} is a separation of π ∈ O(ℓ), we have ℓ(π) = ∑_{i=1}^{m} ℓ(π^{i}), p_{π}(x) = ∏_{i=1}^{m} p_{π_{i}}(x), and using (4.20) and comparing the coefficients of p_{π}(x) on both sides of the above, we obtain the desired formula. □

Remark 1: We note that simplifications of the above formula can be obtained in most cases; that is, in case (i)

2^{\frac{1}{2} (−ϵ+∑_{i∈I} ϵ_{i}^{σ})} = 1

in the following cases:

(a) if ℓ is even and the parts of λ are all even or are all odd,

(b) if ℓ is odd and all the parts of λ except one are even or are odd,

and in case (ii)

2^{\frac{1}{2} (ϵ+ϵ_{σ(1)}+∑_{j∈J} ϵ_{j}^{σ})} = 1

in the following cases

(a) if ℓ is even and all the parts of λ are either even or are odd except λ_{σ(1)},

(b) if ℓ is odd and all the parts of λ are either odd or are even except for λ_{σ(1)}.

Furthermore, in order to apply these formulae, we require to determine left coset representatives of S_{2} ≀ S_{μ} in S_{m} for both m = 2μ and 2μ + 1. We note that

[S_{m} : S_{2} ≀ S_{μ}] = \{ \begin{matrix}1 · 3 · . . . .m − 1 & if m = 2μ is even, \\ 1 · 3 · . . . · m & if m = 2μ + 1 is odd. \end{matrix}

If m = 2μ is even, left coset representatives of S_{2} ≀ S_{μ} in S_{m} are of the form ψ = x_{1}x_{2} … x_{m−1}, where x_{i} = (x_{i1}, x_{i2}) are transpositions determined inductively as follows:

(i) Put I_{1} = {2, …, m}, then x_{11} = 2, x_{12} = k_{1}, k_{1} ∈ I_{1}.

(ii) Put I_{2} = I \ {x_{11}, x_{12}}, d_{2} = min I_{2}, then x_{21} = d_{2}, x_{22} = k_{2}, k_{2} ∈ I_{2} unless k_{1} = 2, in which case x_{21} = d_{2} + 1.

(iii) Put I_{r} = I_{r−1} \ {x_{r−1,1}, x_{r−1,2}}, d_{r} = min I_{r}, then x_{r1} = d_{r}, x_{r2} = k_{r}, k_{r} ∈ I_{r}, unless k_{r−1} = d_{r−1}, in which case x_{r2} = d_{r−1} + 1.

For example, the left coset representatives of S_{2} ≀ S_{3} in S_{6} are given as follows

[figure: (44); (45); (46); (44); (45); (46); (33); (35); (36); (33); (34); (36); (33); (34); (36); (22); (23); (24); (25); (26); e]

that is, {e, (45), (46), (23), (23)(45), (23)(46), (24), (24)(35), (24)(36), (25), (25)(34), (25)(36), (26), (26)(34), (26)(35)}.

Similarly, if m = 2μ + 1 is odd, the left coset representatives of S_{2} ≀ S_{μ} in S_{m} are of the form ϕ = ϕ_{1}ψ, where ϕ_{1} can be any of the m transpositions (1, k) (1 ≤ k ≤ m) and ψ is obtained in exactly the same way as the above but now applied to the set J = {3, 4, …, m = 2μ + 1} in place of I = {2, …, 2μ}. Thus, for example, the left coset representatives of S_{2} ≀ S_{2} in S_{5} are {e, (12), (13), (14), (15)}×{e, (34), (35)} = {e, (34), (35), 12), (12)(34), (12)(35), (13), (13)(34), (13)(35),(14), (14)(34), (14)(35), (15), (15)(34), (15)(35)}. Example 1: The values of the character ζ^{(6,4,3,2)} of S_{15} are calculated at the classes (1^{2}3^{2}7) and (3, 5, 7). In this case, as (λ_{1}, λ_{2}, λ_{3}, λ_{4}) = (6, 4, 3, 2), the relevant left coset representatives are e, (23), (24); thus, by Theorem 4.2(i) and Remark 1(i), we have

ζ_{(1^{2}3^{2}7)}^{(6,4,3,2)} = \frac{2!2!}{2!}ζ_{(3,7)}^{(6,4)}ζ_{(1^{2}3)}^{(3,2)} − \frac{2!2!}{2!2!}ζ_{(1^{2}7)}^{(6,3)}ζ_{(3^{2})}^{(4,2)}

+\frac{2!2!}{2!}ζ_{(1,7)}^{(6,2)}ζ_{(1,3^{2})}^{(4,3)} + \frac{2!2!}{2!2!}ζ_{(1^{2}3^{2})}^{(6,2)}ζ_{(7)}^{(4,3)},

noting that ((3, 7), (1^{2}3, ϕ)) and ((1^{2}7, ϕ), (3^{2}, ϕ)) are the only separations of (1^{2}3^{2}7) corresponding to ((6, 4), (3, 2)) and ((6, 3), (4, 2)) respectively, while there are two separations ((1, 7), (13^{2}, ϕ)) and ((1^{2}3^{2}), ϕ), ((7), ϕ) of (1^{2}3^{2}7) corresponding to ((6, 2), (4, 3, 1)).

Now by means of Theorem 4.1, we find that

ζ_{(3,7)}^{(6,4)} = 0, ζ_{(1^{2},3)}^{(3,2)} = −1, ζ_{(1^{2}7)}^{(6,3)} = 0, ζ_{(3^{2})}^{(4,2)} = 2, ζ_{(17)}^{(6,2)} = 0,
ζ_{(13^{2})}^{(4,3)} = 2, ζ_{(1^{2}3^{2})}^{(6,2)} = −2, ζ_{(7)}^{(4,3)} = −1.

Thus

ζ_{(1^{2}3^{2}7)}^{(6,4,3,2)} = ζ_{(1^{2}3^{2})}^{(6,2)}ζ_{(7)}^{(4,3)} = 2.

Since the partition (3, 5, 7) has no separations corresponding to π_{9} ∈ O(9), π_{6} ∈ O(6), we have

ζ_{(3,5,7)}^{(6,4,3,2)} = ζ_{(3,7)}^{(6,4)}ζ_{(5)}^{(3,2)} + ζ_{(3,5)}^{(6,2)}ζ_{(7)}^{(4,3)}
= 0 · 1 − 2 · (−1) = +2.

Example 2: The values of the character ζ^{(5,4,3,2,1)} of S_{15} are calculated at the classes (1^{2}3^{2}7) and (3, 5, 7). In this case, there are terms corresponding to the 15 left coset representatives of S_{2} ≀ S_{2} in S_{5} listed above. However, since ζ_{(1,7)}^{(5,3)} = ζ_{(1^{2}7)}^{(5,4)} = ζ_{(1^{2}3)}^{(4,1)} = ζ_{(13^{2}}^{(5,2)} = 0, then by Theorem 4.2(ii) and Remark 1(ii) and taking the only possible separations of (1^{2}3^{2}7) we obtain

ζ_{(1^{2}3^{2}7)}^{(5,4,3,2,1)} = \frac{2!2!}{2!} · 2ζ_{(1^{2}3)}^{(5)}ζ_{(7)}^{(4,3)}ζ_{(3)}^{(2,1)} + \frac{2!2!}{1} · 2 · ζ_{(1,3)}^{(4)}ζ_{(7)}^{(5,2)}ζ_{(1,3)}^{(3,1)}

−(−\frac{2!2!}{2!2!} · 2ζ_{(1^{2})}^{(2)}ζ_{(1^{2})}^{(2)}ζ_{(3^{2})}^{(5,1)}ζ_{(7)}^{(4,3)}) + \frac{2!2!}{2!} · 2 · ζ_{(1)}^{(1)}ζ_{(7)}^{(5,2)}ζ_{(13^{2})}^{(4,3)}.

Again, by Theorem 4.1, we have ζ_{(7)}^{(4,3)} = −1, ζ_{(3)}^{(2,1)} = −1, ζ_{(7)}^{(5,2)} = 1, ζ_{(1,3)}^{(3,1)} = −(−1), ζ_{(3^{2})}^{(5,1)} = −2 and ζ_{(13^{2})}^{(4,3)} = 2, and by (4.17), we have ζ_{(1^{2}3)}^{(5)} = 2, ζ_{(13)}^{(4)} = 1, ζ_{(1^{2})}^{(2)} = 1 and ζ_{(1)}^{(1)} = 1. Thus, we have

ζ_{(1^{2}3^{2}7)}^{(5,4,3,2,1)} = 4.

Due to the separation process, there are only two terms to be considered for the class (3, 5, 7) and we obtain

\begin{aligned}ζ_{(3,5,7)}^{(5,4,3,2,1)} & = 2ζ_{(5)}^{(5)}ζ_{(7)}^{(4,3)}ζ_{(3)}^{(2,1)} − 2ζ_{(3)}^{(3)}ζ_{(7)}^{(5,2)}ζ_{(5)}^{(4,1)} \\ & = 2 · 1 · (−1) · (−1) − 2 · 1 · 1 · (−1) = 4. \end{aligned}

References

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[2] G.D. James and A. Kerber: The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, Mass., 1981.

[3] D.E. Littlewood and A.R. Richardson: Group characters and algebras, Phil. Trans. Roy. Soc. London, Ser A 233 (1934).

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