Spin representations, powers of 2 and the Glaisher map

Christine Bessenrodt

Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg

D-39016 Magdeburg, Germany Email: bessen@mathematik.uni-magdeburg.de

Jørn B. Olsson

Matematisk Afdeling, University of Copenhagen Copenhagen, Denmark Email: olsson@math.ku.dk

December 14, 2001

Abstract

Generalizing results on spin character degrees, we determine for a given conjugacy class of odd type in the double cover of S_{n} spin characters of S_{n} which have the minimal 2-power on this class in their character value. Surprisingly, the Glaisher map plays an important rôle here. Mathematics Subject Classification (2000): 20C30, 05A15

1 Introduction

In this article we consider the values of spin characters of a double cover Ŝ_{n} of the symmetric group S_{n} on 2-regular classes. In [BO] we have determined the maximal 2-power dividing all spin character values on a fixed conjugacy class corresponding to a cycle type with odd parts only. For the class (1^{n}), i.e., the spin character degrees, Wagner [W] showed that the spin character corresponding to the 2-adic decomposition of n has the minimal 2-power in its degree. This was also obtained in [BO97], where we also determined explicitly all the spin characters where this minimum is attained. In this note, we are now generalizing and refining the results on the 2-powers in the spin character degrees to all odd classes.

Before we can state the main result precisely, we need to introduce some notation. The associate classes of spin characters of Ŝ_{n} are labelled canonically by the partitions λ of n into distinct parts, i.e. λ = (λ_{1}, λ_{2}, ⋯, λ_{m}), λ_{1} > λ_{2} > ⋯ > λ_{m} > 0, λ_{1} + ⋯ + λ_{m} = n. We write |λ| = n and ℓ(λ) = m, the cardinality and length of λ. Also the sign of λ is sgn(λ) = (−1)^{n−m}. According to the signs the set D(n) of partitions of n into distinct parts is divided into disjoint subsets D^{+}(n) and D^{−}(n).

Then the self-associate spin characters in S_{n} are labelled by the partitions in D^{+}(n) and the associate pairs of spin characters are labelled by the partitions in D^{−}(n).

The conjugacy classes of elements of odd order in S_{n} are labelled canonically via their cycle type by the elements in the set O(n) of partitions of n into odd parts. We will use an ‘exponential’ notation for partitions α ∈ O(n):

α = (1^{m_{1}}, 3^{m_{3}}, ⋯). Thus |α| = ∑_{i odd} i m_{i} , ℓ(α) = ∑_{i odd} m_{i}.

It is well known that |D(n)| = |O(n)|; we denote this cardinality by d(n). In fact, J.W.L. Glaisher [G] defined a bijection between partitions with parts not divisible by a given number k on the one hand and partitions where no part is repeated k times on the other hand; so in particular for k = 2 this gives a bijection between O(n) and D(n). In this situation, Glaisher’s map G is defined as follows. Suppose that α = (1^{m_{1}}, 3^{m_{3}}, ⋯) ∈ O(n). Write each multiplicity m_{i} as a sum of distinct powers of 2, i.e., in its 2adic decomposition: m_{i} = ∑_{j} 2^{a_{ij}}. Then G(α) ∈ D(n) consists of the parts (2^{a_{ij}}i)_{i,j}, of course in descending order.

For any integer m ≥ 0, let s(m) be the number of summands in the 2adic decomposition of m. Then for α = (1^{m_{1}}, 3^{m_{3}}, ⋯) the length of G(α) is ℓ(G(α)) = ∑_{i odd} s(m_{i}). We define k_{α} = ∑_{i odd}(m_{i} − s(m_{i})) and set σ(α) =

(−1)^{k_{α}}. We denote by O^{ε}(n) the set of partitions α in O(n) with the sign of σ(α) being ε. Then the Glaisher map G induces bijections O^{ϵ}(n) → D^{ϵ}(n), where ϵ is a sign (see [BO]).

The integer k_{α} is also of group-theoretic significance. For any integer m, we denote by ν(m) the exponent to which 2 divides m. Thus m_{2} := 2^{ν(m)} is the exact 2-power dividing m. Let α ∈ O(n) and let x′_{α} be an element of cycle type α in S_{n}. Then ν(|C_{S_{n}}(x′_{α})|) = ∏_{i odd} ν(m_{i}!) = k_{α}. So k_{α} is the

2-defect of C_{α}, the conjugacy class of S_{n} labelled by α ∈ O(n).

The preimage π^{−1}(C(α)) consists of two conjugacy classes in Ŝ_{n}, say C_{α}^{(1)} and C_{α}^{(2)}. We choose notation such that the elements of C_{α}^{(1)} have odd order, and we let x_{α} be the preimage of x′_{α} in this class. Then C_{α}^{(2)} = z C_{α}^{(1)}, and the elements in this second conjugacy class have even order. These conjugacy classes have 2-defect k_{α} + 1.

The result from [BO] mentioned above says:

Theorem 1.1 Let α = (1^{m_{1}}, 3^{m_{3}}, ⋯) ∈ O(n). Then 2^{[k_{α}/2]} is the maximal power of 2 which divides all spin character values ⟨λ⟩(x_{α}), λ ∈ D(n), i.e., ν(⟨λ⟩(x_{α})) ≥ [\frac{k_{α}}{2} ] for all λ ∈ D(n).

The goal now is to prove

Theorem 1.2 Let α = (1^{m_{1}}, 3^{m_{3}}, ⋯) ∈ O(n), and let λ = G(α) be its Glaisher image. Then ν(⟨λ⟩(x_{α})) = [\frac{k_{α}}{2} ]. Furthermore, if λ ∈ D^{−}(n), then ⟨λ⟩ and ⟨λ⟩′ are the only spin characters where this equality holds.

2 Preliminaries

Here we collect the results that are needed later on.

From [BO97] we need the description of the set of spin characters of s.a./n.s.a. type where the minimal 2-power in the degree is attained.

For n ∈ ℕ we set

M̄(n) = \{λ ∈ D(n) | ν(⟨λ⟩(1)) = [\frac{n − s(n)}{2}]\}

M̄^{+}(n) = \{λ ∈ D^{+}(n) | ν(⟨λ⟩(1)) = [\frac{n − s(n) + 1}{2}]\}

M̄^{−}(n) = \{λ ∈ D^{−}(n) | ν(⟨λ⟩(1)) = [\frac{n − s(n)}{2}]\} = M̄(n) ∩ D^{−}(n)

M̄_{1}(n) = \{λ ∈ D(n) | ν(⟨λ⟩(1)) = [\frac{n − s(n)}{2}] + 1\}

Furthermore, if n = ∑_{i=1}^{s} 2^{k_{i}}, k_{1} > k_{2} … > k_{s}, is the 2-adic decomposition of n, then we denote by δ(n) = (2^{k_{1}}, 2^{k_{2}}, …, 2^{k_{s}}) the corresponding partition of n. So sgn(δ(n)) = σ(1^{n}).

If λ ∈ D, we write λ = ∑_{i≥0} 2^{i}λ_{i} with λ_{i} ∈ D ∩ O for all i. For any partition ρ let s(ρ) = s(|ρ|).

A careful analysis of the proof of part (b) of Theorem 2.6 in [BO97] shows that the description given in Theorem 2.7 [BO97] can be improved a little bit:

Theorem 2.1 ([BO97]) Let n ∈ ℕ, s = s(n), and let ε be a sign. Set

D_{0}^{ε}(n) = {λ = ∑_{i≥0} 2^{i}λ_{i} ∈ D^{ε}(n) | ∃!i_{0} : |λ_{i_{0}}| > 1; and this λ_{i_{0}} satisfies:
s(λ_{i_{0}}) ≤ 2, λ_{i_{0}} ∈ M̄_{1}, s = |{λ_{i} ≠ ∅}| + s(λ_{i_{0}}) − 1}

Then

M̄^{ε}(n) = \{ \begin{matrix}\{δ(n)\} & if ε = sgn(δ(n)) \\ D_{0}^{ε}(n) & otherwise \end{matrix}

M̄(n) = \{ \begin{matrix}\{δ(n)\} ∪ D_{0}^{−}(n) & if σ(1^{n}) = +1 \\ \{δ(n)\} & if σ(1^{n}) = −1 \end{matrix}

From [MO] we need the values of spin characters on a class of type (e^{m}), e odd. For a partition λ we denote by H(λ) the product of its hook lengths, and if λ ∈ D we denote by H̄(λ) the product of its bar lengths.

Theorem 2.2 ([MO], Cor. (4.3)) Let ρ ∈ D(me), e odd, with ρ_{(ē)} = ∅, ρ^{(ē)} = (ρ_{0}; ρ_{1}, …, ρ_{t}), t = (e − 1)/2. Then

⟨ρ⟩(x_{(e^{m})}) = ±2^{[ \frac{m−ℓ(ρ_{0})}{2} ]} \frac{m!}{H̄(ρ^{(ē)})}

where H̄(ρ^{(ē)}) = H̄(ρ_{0}) ∏_{j=1}^{t} H(ρ_{j}).

We can rewrite this in the following form:

Corollary 2.3 Set r = [\frac{m−ℓ(ρ_{0})}{2}] − [\frac{|ρ_{0}|−ℓ(ρ_{0})}{2}]. Then

⟨ρ⟩(x_{(e^{m})}) = ±2^{r}\binom{m}{|ρ_{0}|, …, |ρ_{t}|}⟨ρ_{0}⟩(1)[ρ_{1}](1) ⋯ [ρ_{t}](1)

Remark. Since ν(n) = n − s(n) we have

ν(\binom{m}{|ρ_{0}|, …, |ρ_{t}|}) = ∑_{j=0}^{t} s(ρ_{j}) − s(m) .

For a reduction to the analogues of Young subgroups we have to collect some results on reduced Clifford products (see [H], [MiO], [S]).

Theorem 2.4 Let χ_{i}, ψ_{i} be spin characters of Ŝ_{a_{i}}, i = 1, …, k. Then the reduced Clifford products χ = ×_{c}χ_{i} and ψ = ×_{c}ψ_{i} have the following properties.

(i) χ is n.s.a. if and only if t_{χ} = |{i | χ_{i} n.s.a.}| is odd.

(ii) Let α^{i} be a partition of a_{i}, i = 1, … k, α = (α^{1}, α^{2}, …).

Then χ(x_{α}) = 2^{[t_{μ}/2]} ∏_{i} χ_{i}(x_{α^{i}}).

(iii) χ and ψ are associate if and only if χ_{i} and ψ_{i} are associate for all i, and they are equal if and only if they are associate and t_{χ} is even, or t_{χ} is odd but |{i | χ_{i} ≠ ψ_{i}}| is even.

3 Proof of the Theorem

In generalizing our previous result for spin character degrees we first deal with the case of a class of type α = (e^{m}), e odd. Then the minimal 2-value of the spin character values on this class is [\frac{k_{α}}{2} ] = [\frac{m−s(m)}{2}], and we now characterize the spin characters where this minimum is attained:

Proposition 3.1 Let ρ ∈ D(me), e odd. Set v_{ρ} = ⟨ρ⟩(x_{(e^{m})}). Let ρ^{(ē)} = (ρ_{0}; ρ_{1}, …, ρ_{t}), t = (e − 1)/2. Then ν(v_{ρ}) = [\frac{m−s(m)}{2}] if and only if ρ_{(ē)} = ∅ and one of the following holds:

(i) ρ^{(ē)} = (ρ_{0}; ∅, …, ∅) and ρ_{0} ∈ M̄(m).

(In particular, ρ = G(e^{m}) is of this type.)

(ii) m − s(m) is even, ρ_{0} ∈ M̄(|ρ_{0}|), s(ρ_{0}) + 1 = s(m), there is a unique l > 0 such that ρ_{l} ≠ ∅, and |ρ_{l}| is a 2-power, [ρ_{l}](1) is odd, and either (a) ℓ(ρ_{0}) = s(ρ_{0}), or (b) ℓ(ρ_{0}) ≠ s(ρ_{0}) and |ρ_{0}| − s(ρ_{0}) odd.

We have ρ ∈ D^{−}(me) in case (i) when ρ_{0} ≠ δ(m), and in case (ii).

Proof. We already know from [BO] that the minimal 2-value on x_{α} is [k_{α}/2], so in our case we deduce that ν(v_{ρ}) ≥ [(m − s(m)/2]. If ρ_{(ē)} ≠ ∅, then v_{ρ} = 0, so these partitions never have minimal 2-value on (e^{m}). Hence we may assume from now on that ρ_{(ē)} = ∅.

Assume first that ρ^{(ē)} = (ρ_{0}; ∅, …, ∅). Then by 2.3, ν(v_{ρ}) = [(m − s(m))/2] if and only if ρ_{0} ∈ M̄(m). This is case (i) in the Proposition.

So we assume from now on that |ρ_{0}| < m. Set s_{j} = s(ρ_{j}); then the assumption is equivalent to saying that ∑_{j>0} s_{j} ≥ 1.

Case: ρ_{0} ∈ M̄(|ρ_{0}|). If ℓ(ρ_{0}) = s_{0}, then we have (by Corollary 2.3 and the remark following it)

\begin{aligned}ν(v_{ρ}) & = & [\frac{m − s_{0}}{2} ] + ∑_{j≥0} s_{j} − s(m) + ν(∏_{j>0}[ρ_{j}](1)) \\ & = & [\frac{m − s(m)}{2} + \frac{1}{2}(∑_{j≥0} s_{j} − s(m)) + \frac{1}{2}(∑_{j>0} s_{j})] + ν(∏_{j>0}[ρ_{j}](1)) \end{aligned}

Since ∑_{j≥0} s_{j} − s(m) ≥ 0 and ∑_{j>0} s_{j} > 0, we have ν(v_{ρ}) = [(m − s(m))/2] if and only if we are in the situation described in case (ii)(a) in the statement of the Proposition.

Now we deal with the case where l_{0} := ℓ(ρ_{0}) ≠ s_{0}. In particular, then ρ_{0} ∈ D^{−} (because of the description of the set M̄). In this case we have (again using Corollary 2.3)

ν(v_{ρ}) = [\frac{m − l_{0}}{2} ] − [\frac{|ρ_{0}| − l_{0}}{2} ] + ∑_{j≥0} s_{j} − s(m) + [\frac{|ρ_{0}| − s_{0}}{2} ] + ν(∏_{j>0}[ρ_{j}](1))

= [\frac{m − s(m)}{2} − \frac{|ρ_{0}| − 1}{2} + \frac{1}{2}(∑_{j≥0} s_{j} − s(m)) + \frac{1}{2}(∑_{j≥0} s_{j})] + [\frac{|ρ_{0}| − s_{0}}{2} ]

+ν(∏_{j>0}[ρ_{j}](1))

If |ρ_{0}| − s_{0} is even, this simplifies to

ν(v_{ρ}) = [\frac{m − s(m)}{2} + \frac{1}{2} + \frac{1}{2}(∑_{j≥0} s_{j} − s(m)) + \frac{1}{2}(∑_{j>0} s_{j})] + ν(∏_{j>0}[ρ_{j}](1))

and this is always larger than [(m − s(m))/2]. If |ρ_{0}| − s_{0} is odd, the expression above simplifies to

ν(v_{ρ}) = [\frac{m − s(m)}{2} + \frac{1}{2}(∑_{j≥0} s_{j} − s(m)) + \frac{1}{2}(∑_{j>0} s_{j})] + ν(∏_{j>0}[ρ_{j}](1))

and this is equal to [(m−s(m))/2] exactly if we are in the situation described in case (ii)(b) in the statement of the Proposition.

Case: ρ_{0} ∉ M̄(|ρ_{0}|). Then

ν(v_{ρ}) ≥ [\frac{m − l_{0}}{2} ] − [\frac{|ρ_{0}| − l_{0}}{2} ] + ∑_{j≥0} s_{j} − s(m) + [\frac{|ρ_{0}| − s_{0}}{2} ] + 1

≥ [\frac{m − s(m)}{2} + \frac{1}{2}(∑_{j≥0} s_{j} − s(m)) + \frac{1}{2}(∑_{j>0} s_{j})] + \frac{1}{2}
> [\frac{m − s(m)}{2}]

So here we do not have any further partitions of minimal 2-value on x_{ρ}.

If in case (i) ρ_{0} ≠ δ(m), then we are in the situation where m ≡ s(m) (mod 2), δ(m) ∈ D^{+} and ρ_{0} ∈ D^{−}. But then clearly, ρ ∈ D^{−}(me). It remains to check that ρ ∈ D^{−}(me) in case (ii). In case (ii)(a), we have

me − ℓ(ρ) ≡ m − ℓ(ρ_{0}) = m − (s(m) − 1) ≡ 1 (mod 2)

In case (ii)(b), we had already seen that ρ_{0} ∈ D^{−}, so

me − ℓ(ρ) ≡ m − ℓ(ρ_{0}) ≡ m − s(ρ_{0}) = m − (s(m) − 1) ≡ 1 (mod 2)

Hence in both situations ρ_{0} ∈ D^{−}. ⋄

Remark. For m = 1, e odd, we obtain: the spin characters ⟨e⟩, ⟨e − 1, 1⟩, ⟨e − 2, 2⟩, …, ⟨\frac{e+1}{2}, \frac{e−1}{2}⟩ (and their associates) have odd value on x_{(e)}. In fact, these values are ±1, and all other spin character values on this class are zero.

We denote the set of those partitions ρ for which the corresponding spin character attains the minimal 2-value on x_{(e^{m})} by M̄(e^{m}). Then we have Corollary 3.2 Let e, m ∈ ℕ, e odd; let (e^{m}) ∈ O^{ε}(em). Then

M̄(e^{m}) ∩ D^{ε} = {G(e^{m})}

and if ε = −, then

M̄(e^{m}) = {G(e^{m})} .

As in the case of the spin degrees, also the following set plays a special rôle:

M̄^{+}(e^{m}) = \{ρ ∈ D^{+}(me) | ⟨ρ⟩(x_{(e^{m})}) = [\frac{m − s(m) + 1}{2}]\}

For reduction purposes we need the following result.

Proposition 3.3 Let α = (1^{m_{1}}, 3^{m_{3}}, …) ∈ O(n). Set α^{i} = (i^{m_{i}}), a_{i} = im_{i}, i = 1, 3, …, and let S̃_{a} be the preimage of the Young subgroup S_{a_{1}} ×S_{a_{3}} ×… in S̃_{n}.

Then

ν((×_{c}⟨μ_{i}⟩)(x_{α})) ≥ [\frac{k_{α}}{2} ]

for all μ = (μ_{1}, μ_{3}, …), μ_{i} ∈ D(a_{i}), and equality is attained on the set M^{∗}(α) described as follows. Let g(α) be the partition sequence defined by g(α)_{i} = G(i^{m_{i}}) for all i. If α ∈ O^{−}, then

M^{∗}(α) = {g(α)} .

If α ∈ O^{+}, then

M^{∗}(α) = {g(α)} ∪ {μ | ∃!l : μ_{l} ∈ M̄(l^{m_{l}}), sgn(μ_{l}) ≠ σ(l^{m_{l}}),
or μ_{l} ∈ M̄^{+}(l^{m_{l}}), σ(l^{m_{l}}) = −1,
and μ_{i} = G(i^{m_{i}}) for all i ≠ l} .

Proof. For a sign ε = ±, let J_{α}^{ε} = {i | α^{i} ∈ O^{ε}}. For a given μ, set T_{μ}^{−} = {i | μ_{i} ∈ D^{−}}, I_{0}^{ε}(μ) = {i ∈ J_{α}^{ε} | μ_{i} ∈ M̄(α^{i})} and I_{1}^{ε}(μ) = J_{α}^{ε} \ I_{0}^{ε}(μ). Let t_{μ} = |T_{μ}^{−}| and set

c_{μ} = (×_{c}⟨μ_{i}⟩)(x_{α}) = 2^{[^{\frac{tμ}{2}} ]} ∏_{i}⟨μ_{i}⟩(x_{α^{i}}) .

For i ∈ I_{1}^{ε}(μ) let r_{i} = ν(⟨μ_{i}⟩(x_{α^{i}})) − [\frac{m_{i}−s_{i}}{2} ] ∈ ℕ. Then

ν(c_{μ}) = [\frac{t_{μ}}{2} ] + ∑_{i∈I_{0}^{+}(μ)} \frac{m_{i} − s_{i}}{2} + ∑_{i∈I_{1}^{+}(μ)} (\frac{m_{i} − s_{i}}{2} + r_{i}) +

+ ∑_{i∈I_{0}^{−}(μ)} \frac{m_{i} − s_{i} − 1}{2} + ∑_{i∈I_{1}^{−}(μ)} (\frac{m_{i} − s_{i} − 1}{2} + r_{i})

= [\frac{k_{α}}{2} + \frac{1}{2}(t_{μ} − |I_{0}^{−}(μ)|) + ∑_{i∈I_{1}^{+}(μ)} r_{i} + ∑_{i∈I_{1}^{−}(μ)} (r_{i} − \frac{1}{2})]

Since T_{μ}^{−} ⊇ I_{0}^{−}(μ), this shows that ν(c_{μ}) ≥ [\frac{k_{α}}{2} ], and that equality holds if and only if I_{1}^{+}(μ) = ∅ and either T_{μ}^{−} = I_{0}^{−}(μ), I_{1}^{−}(μ) = ∅ or T_{μ}^{−} = I_{0}^{−}(μ), I_{1}^{−}(μ) = {l} for some l with r_{l} = 1, or T_{μ}^{−} = I_{0}^{−}(μ) ∪ {l} for some l ∈ I_{0}^{+}(μ). Together with our previous results, this gives the description of the set M^{∗}(α) stated in the Proposition. ⋄

Proof of Theorem 1.2. Let α = (i^{m_{i}})_{i=1,3,...} ∈ O, α^{i} = (i^{m_{i}}) ⊢ a_{i} = im_{i}, S̃_{a} as before.

Let λ ∈ D. Restricting to S̃_{a} gives

⟨λ⟩S̃_{a} = ∑_{μ=(μ_{1},μ_{3},...)} g_{μ}^{λ}(×_{c}⟨μ_{i}⟩) + ∑_{μ=(μ_{1},μ_{3},...)n.s.a.} ḡ_{μ}^{λ}(×_{c}μ_{i})′

where μ = (μ_{1}, μ_{3}, …) runs over all partition sequences with μ_{i} ⊢ a_{i}, and we call μ n.s.a. if the corresponding reduced Clifford product is n.s.a.

Then

⟨λ⟩(x_{α}) = ∑_{μ} g_{μ}^{λ} 2^{[t_{μ}/2]} ∏_{i}⟨μ_{i}⟩(x_{(i^{m_{i}})}) + ∑_{μ n.s.a.} ḡ_{μ}^{λ} 2^{[t_{μ}/2]} ∏_{i}⟨μ_{i}⟩(x_{(i^{m_{i}})})

As the 2-value of each summand is at least [k_{α}/2] by Proposition 3.3, ´ we can have equality only if there exists μ ∈ M^{∗}(α) such that g_{μ}^{λ} is odd. Furthermore, note that by [St] we have for n.s.a. μ: g_{μ}^{λ} = ḡ_{μ}^{λ} whenever λ ∈ D^{+}, or λ ∈ D^{−} but λ ≠ ⋃ μ_{i}. So taking the associate characters together, we get for the n.s.a. μ a contribution of non-minimal 2-value for such λ.

Case: α ∈ O^{−}. Let λ = G(α). We want to show that ⟨λ⟩ and its associate are the only spin characters which have the minimal 2-value [k_{α}/2] on x_{α}. Since α ∈ O^{−}, by Proposition 3.3 the set M^{∗}(α) only contains the n.s.a. partition sequence μ = g(α). By the considerations above, the only possible ⟨λ⟩ with minimal 2-value on x_{α} then is given by λ = ⋃_{i} μ_{i} = G(α), as was to be proved.

Case: α ∈ O^{+}. Let λ = G(α). We want to show that ⟨λ⟩ has the minimal 2-value [k_{α}/2] on x_{α}. By Proposition 3.3, for α ∈ O^{+} the set M^{∗}(α) contains two types of partition sequences. On the one hand, we have the s.a. partition sequence μ = g(α). On the other hand, we have certain n.s.a. μ ∈ M^{∗}(α), but as remarked above, together with the associate character this gives a summand of non-minimal 2-value for ⟨λ⟩(x_{α}). So it only remains to show that the spin Littlewood-Richardson coefficient g_{g(α)}^{G(α)}, i.e., the multiplicity of ⟨G(α)⟩ in the induction to Ŝ_{n} of the reduced Clifford product corresponding to g(α), is odd. Indeed, g_{g(α)}^{G(α)} = 1 by the following result:

Lemma 3.4 Let ν = (ν^{1}, ν^{2}, …) be a sequence of partitions ν^{i} ∈ D, with pairwise disjoint parts. Let ρ = ⋃_{i} ν^{i}, |ρ| = n. Then

g_{ν}^{ρ} = ((×_{c}⟨ν^{i}⟩) ↑S̃_{n}, ⟨ρ⟩) = 1 .

Proof. Use the spin analogue of the Littlewood-Richardson rule due to Stembridge [St]. ⋄

Remarks. More precisely, we believe that also in the second case, when α ∈ O^{+}, the spin character ⟨G(α)⟩ is the only one with its sign being equal to σ(α) that is minimal on x_{α}. The only critical candidates are λ ∈ D^{+} such that g_{g(α)}^{λ} is odd; in particular, then we must have ℓ(λ) = ℓ(G(α)). It seems to require a rather delicate tableaux counting argument to show that this situation cannot occur.

Moreover, in this second case there usually do exist n.s.a. spin characters that are minimal on x_{α}. These come from the n.s.a. partition sequences in M^{∗}(α) (by taking unions of the partitions in the sequence); note that we then have a choice different from the Glaisher image at the component μ_{i} where the sign differs from that of the corresponding α^{i}. These spin characters are still labelled by partitions ‘close’ to the Glaisher image of α.

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