Properties of some character tables related to the symmetric groups

Christine Bessenrodt, Jørn B. Olsson^{1}, Richard P. Stanley^{2}

Abstract We determine invariants like the Smith normal form and the determinant for certain integral matrices which arise from the character tables of the symmetric groups S_{n} and their double covers. In particular, we give a simple computation, based on the theory of Hall-Littlewood symmetric functions, of the determinant of the regular character table X_{RC} of S_{n} with respect to an integer r ≥ 2. This result had earlier been proved by Olsson in a longer and more indirect manner. As a consequence, we obtain a new proof of the Mathas’ Conjecture on the determinant of the Cartan matrix of the Iwahori-Hecke algebra. When r is prime we determine the Smith normal form of X_{RC}. Taking r large yields the Smith normal form of the full character table of S_{n}. Analogous results are then given for spin characters.

1 Introduction

In this paper we determine invariants like the Smith normal form or the determinant for certain integral matrices which come from the character tables of the finite symmetric groups S_{n} and their double covers Ŝ_{n}. The matrices in question are the so-called regular and singular character tables of S_{n} and the reduced spin character table of Ŝ_{n}.

In section 2 we calculate the determinants of the r-regular and r-singular character tables of S_{n} for arbitrary integers r ≥ 2, using symmetric functions and some bijections involving regular partitions. The knowledge of these determinants is equivalent to the knowledge of the determinants of certain “generalized Cartan matrices” of S_{n} as considered in [9]. In particular we obtain a new proof of a conjecture of Mathas about the Cartan matrix of an Iwahori-Hecke algebra of S_{n} at a primitive rth root of unity which is simpler than the original proof given by Brundan and Kleshchev in [3]. In section 3 we determine the Smith normal form of the regular character table in the case where r is a prime. As a special case the Smith normal form of the character table of S_{n} may be calculated. We also determine the Smith normal form of the reduced spin character table for Ŝ_{n}. The paper also presents some open questions.

2 The determinant of the regular part of the character table of S_{n}

We fix positive integers n, r, where r ≥ 2.

If μ = (μ_{1}, μ_{2}, …) is a partition of n we write μ ∈ P and denote by ℓ(μ) the number of (non-zero) parts of μ. We let z_{μ} denote the order of the centralizer of an element of (conjugacy) type μ in S_{n}. Suppose μ = (1^{m_{1}(μ)}, 2^{m_{2}(μ)}, …), is written in exponential notation. Then we may factor z_{μ} = a_{μ}b_{μ}, where

a_{μ} = ∏_{i≥1} i^{m_{i}(μ)}, b_{μ} = ∏_{i≥1} m_{i}(μ)!

Whenever Q ⊆ P we define

a_{Q} = ∏_{μ∈Q} a_{μ}, b_{Q} = ∏_{μ∈Q} b_{μ}.

Let μ ∈ P. We write μ ∈ R and call μ regular if m_{i}(μ) ≤ r − 1 for all i ≥ 1. We write μ ∈ C and call μ class regular if m_{i}(μ) = 0, whenever r | i.

We are particularly interested in the integers a_{C} and b_{C}. By [12, Theorem 4] there is a connection between a_{C} and b_{C} given by

b_{C} = r^{d_{C}} a_{C},  (1)

where the class regular defect number d_{C} is defined by

d_{C} = ∑_{μ∈C} d(μ), d(μ) = ∑_{i,k≥1} ⌊\frac{m_{i}(μ)}{r^{k}}⌋ .

Here ⌊·⌋ is the floor function, i.e., ⌊x⌋ denotes the integral part of x. Note that for r > n we have R = C = P and then d_{P} = 0 and thus a_{P} = b_{P}.

Let X_{RC} denote the regular character table of S_{n} with respect to r. It is a submatrix of the character table X of S_{n}. The subscript RC indicates that the rows of X_{RC} are indexed by the set R of regular partitions of n, and the columns by the set C of class regular partitions of n. We want to present a proof of the following result:

Theorem 1 We have

| det (X_{RC}) | = a_{C}.

This result was first proved in [12], but the proof relied on results of [9] for which the work of Donkin [4] and Brundan and Kleshchev [3] was used in a crucial way. Our proof of Theorem 1 does not use [4] or [3]; it is direct and thus much shorter.

In [9], an r-analogue of the modular representation theory for S_{n} was developed systematically, and in particular, an r-analogue of the Cartan matrix for the symmetric groups (and the corresponding r-blocks) was introduced.

In [2] the explicit value of this latter determinant was conjectured to be r^{d_{C}} in the notation above; this was proved in [9, Proposition 6.11] using [4] and [3]. This result is now a consequence of our theorem:

Corollary 2 Let C be the r-analogue of the Cartan matrix of S_{n} as defined in [9]. Then we have

det (C) = r^{d_{C}} .

Proof. As is shown in [12] there is a simple equation connecting the determinants of C and X_{RC}, namely

det(X_{RC})^{2} det(C) = a_{C}b_{C} .

Thus in view of equation (1) Theorem 1 implies the Corollary. □

Mathas conjectured that the determinant of the Cartan matrix of an Iwahori-Hecke algebra of S_{n} at a primitive rth root of unity should be a power of r; via [4], the conjecture in [2] mentioned above predicted the explicit value of this determinant, thus providing a strengthening of Mathas’ conjecture. Mathas’ conjecture was proved by Brundan and Kleshchev [3]; in fact, they also gave an explicit formula for this determinant for blocks of the Hecke algebra. We can now provide an alternative proof of these conjectures.

Corollary 3 The strengthened Mathas’ conjecture is true.

Proof. Donkin [4] has shown that the Cartan matrix for the Hecke algebra has the same determinant as the Cartan matrix C considered in Corollary 2. □

Based on this and the results on r-blocks in [9], the results in [2] then also give the determinants of Cartan matrices of r-blocks of S_{n} explicitly, without the use of [3].

Let us finally mention that in [9, Section 6] there is an explicit conjecture about the Smith normal form of C. In the case where r is a prime, this is known to be true by the general theory of R. Brauer. One may also ask about the Smith normal form of X_{RC}; we answer this question in this article in the prime case.

We now proceed to describe the proof of Theorem 1. It is obtained by combining Theorems 4 and 5 below. Theorem 4 evaluates det (X_{RC})^{2} using symmetric functions as an expression involving a primitive rth root of unity. Theorem 5 shows that this expression equals a_{C}^{2}. It is based on general bijections involving regular partitions.

Define

z_{μ}(t) = z_{μ} ∏_{j}(1 − t^{μ_{j}})^{−1} = z_{μ} ∏_{i}(1 − t^{i})^{−m_{i}(μ)}

where the product ranges over all j for which μ_{j} > 0, and

b_{λ}(t) = ∏_{i}(1 − t)(1 − t^{2}) ⋯ (1 − t^{m_{i}(λ)}).

Let ω = e^{2πi/r}, a primitive rth root of unity.

We use notation from the theory of symmetric functions from [10] or [14]. In particular, m_{λ}, s_{λ}, and p_{λ} denote the monomial, Schur, and power sum symmetric functions, respectively, indexed by the partition λ.

Theorem 4 We have

det (X_{RC})^{2} = ∏_{μ∈C} z_{μ}(ω) · ∏_{λ∈R} b_{λ}(ω).

Proof. Let Q_{λ}(x; t) denote a Hall-Littlewood symmetric function as in [10, p. 210]. It is immediate from the definition of Q_{λ}(x; t) that Q_{λ}(x; ω) = 0 unless λ ∈ R. Moreover (see [10, Exam. III.7.7, p. 249]) when Q_{λ}(x; ω) is expanded in terms of power sums p_{μ}, only class regular μ appear. Thus [10, (7.5), p. 247] for λ ∈ R we have

Q_{λ}(x; ω) = ∑_{μ∈C} z_{μ}(ω)^{−1}X_{μ}^{λ}(ω)p_{μ}(x),

where X_{μ}^{λ}(t) is a Green’s polynomial.

Hence by [10, (7.4)] the matrix X(ω)_{RC} = (X_{μ}^{λ}(ω)), where λ ∈ R and μ ∈ C, satisfies

det(X(ω)_{RC})^{2} = ∏_{μ∈C} z_{μ}(ω) ∏_{λ∈R} b_{λ}(ω).  (2)

Now consider the symmetric function S_{λ}(x; t) as defined in [10, (4.5), p. 224]. It follows from the formula S_{λ}(x; t) = s_{λ}(ξ) in [10, top of p. 225] that

S_{λ}(x; t) = s_{λ}(p_{j} → (1 − t^{j})p_{j}),

i.e., expand s_{λ}(x) as a polynomial in the p_{j}’s and substitute (1 − t^{j})p_{j} for p_{j}. Since

s_{λ} = ∑_{μ} z_{μ}^{−1}χ^{λ}(μ)p_{μ},

we have

S_{λ}(x; ω) = ∑_{μ∈C} z_{μ}(ω)^{−1}χ^{λ}(μ)p_{μ}.

The S_{λ}(x; ω)’s thus lie in the space A_{(r)} spanned over ℚ(ω) by the p_{μ}’s where μ ∈ C. Since the Q_{μ}(x; ω)’s for regular μ span A_{(r)} by [10, Exam. III.7.7, p. 249], the same is true of the S_{λ}(s; ω)’s. Moreover, the transition matrix M(S, Q)_{RR} between the Q_{λ}(x; t)’s and S_{λ}(x; t)’s is lower unitriangular by [10, top of p. 239] and [10, p. 241]. Hence

det M(S, Q)_{RR} = 1.  (3)

Let M(S, p)_{RC} denote the transition matrix from the p_{μ}’s to S_{λ}’s for μ ∈ C and λ ∈ R. Let Z(t)_{CC} denote the diagonal matrix with entries z_{λ}(t), λ ∈ C. By the discussion above we have

X_{RC} = M(S, p)_{RC} Z(ω)_{CC} (by the relevant definitions)
= M(S, Q)_{RR} M(Q, p)_{RC} Z(ω)_{CC}
= M(S, Q)_{RR} X(ω)_{RC} Z(ω)_{CC}^{−1} Z(ω)_{CC}
= M(S, Q)_{RR} X(ω)_{RC}.

Taking determinants and using (2) and (3) completes the proof. □

Define

A_{C}(ω) = ∏_{μ∈C} ∏_{i}(1 − ω^{i})^{−m_{i}(μ)}

B_{R}(ω) = ∏_{λ∈R} b_{λ}(ω)^{−1} = ∏_{λ∈R} (∏_{i}(1 − ω)(1 − ω^{2}) ⋯ (1 − ω^{m_{i}(λ)}))^{−1} ,

so that by Theorem 4

det (X_{RC})^{2} = a_{C}b_{C}A_{C}(ω)B_{R}(ω)^{−1}.

In order to complete the proof of Theorem 1 we thus just need to show:

\frac{B_{R}(ω)}{A_{C}(ω)} = \frac{b_{C}}{a_{C}} .

As \frac{b_{C}}{a_{C}} = r^{d_{C}} this is equivalent to showing Theorem 5 We have

\frac{B_{R}(ω)}{A_{C}(ω)} = r^{d_{C}}.

Clearly the factors 1 − ω^{j} occurring on the left hand side in Theorem 5 depend only on the residue of j modulo r. Thus

A_{C}(ω)^{−1} = ∏_{s=1}^{r−1}(1 − ω^{s})^{α_{C}^{(s)}}, B_{R}(ω)^{−1} = ∏_{s=1}^{r−1}(1 − ω^{s})^{β_{R}^{(s)}},

where

α_{C}^{(s)} = ∑_{μ∈C} ∑_{{i|i≡s(mod r)}} m_{i}(μ)

β_{R}^{(s)} = ∑_{ρ∈R} |{i|m_{i}(ρ) ≥ s}|.

We use the bijections κ^{(s)} defined in Proposition 9 below to show the following:

Proposition 6 For all s ∈ {1, …, r − 1} we have

α_{C}^{(s)} = β_{R}^{(s)} + d_{C}.

This shows then that

\frac{B_{R}(ω)}{A_{C}(ω)} = (∏_{s=1}^{r−1}(1 − ω^{s}))^{d_{C}} .

Then Theorem 5 follows from the fact that

∏_{s=1}^{r−1}(1 − ω^{s}) = r.

(Simply substitute x = 1 in the identity 1 + x + ⋯ + x^{r−1} = ∏_{s=1}^{r−1}(x − ω^{s}).)

Let m ∈ ℕ. We write m in its r-adic decomposition as m = ∑_{j≥0} m_{j}r^{j}, i.e., with m_{j} ∈ {0, …, r − 1} for all j. For m ≠ 0, we can write m = ∑_{j≥k} m_{j}r^{j}, with m_{k} ≠ 0. In the power series convention, k(m) = k is the degree of m and ℓ(m) = m_{k} its leading coefficient. We also set h(m) = ∑_{j≥k+1} m_{j}r^{j} = r^{k+1}q(m) for the higher terms of m. Thus

m = ℓ(m)r^{k(m)} + q(m)r^{k(m)+1} .

For a given a, we define

h_{a}(m) = ∑_{j≥a} m_{j}r^{j} = q_{a}(m)r^{a} , q_{a}(m) = ⌊\frac{m}{r^{a}}⌋ .

We call e ∈ {1, …, m} a non-defect number for m, if h(e) = h_{k(e)+1}(m), otherwise e is a defect number for m (and then h(e) < h_{k(e)+1}(m), and hence q(e) < q_{k(e)+1}(m)). Thus the non-defect numbers for m are of the form

e = e_{a}r^{a} + h_{a+1}(m) , e_{a} ∈ {1, …, m_{a}} ,

and thus there are ∑_{j≥0} m_{j} such numbers. The defect numbers for m are of the form

e = e_{a}r^{a} + qr^{a+1} , e_{a} ∈ {1, …, r − 1} , q ∈ {0, …, q_{a+1}(m) − 1} .

Their parameters (a, q) thus belong to the set

D(m) = {(a, q) | a ≥ 0, 0 ≤ q < q_{a+1}(m)},

which is of cardinality

d(m) = ∑_{a≥1} ⌊\frac{m}{r^{a}}⌋ ,

called the defect of m. For each s ∈ {1, …, r − 1} there are exactly d(m) defect numbers for m with leading coefficient s, namely e = sr^{a} + qr^{a+1}, where (a, q) ∈ D(m). Thus clearly we have (r − 1)d(m) defect numbers for m and

m = (r − 1)d(m) + ∑_{j≥0} m_{j} .

For μ ∈ P, its defect (as defined at the beginning of this section) is then

d(μ) = ∑_{i≥1} d(m_{i}(μ)).

For s ∈ {1, …, r − 1} set

D^{(s)}(μ) = {(i, a, q) | ℓ(i) = s, (a, q) ∈ D(m_{i}(μ))}

and

D(μ) = ⋃_{s=1}^{r−1} D^{(s)}(μ) .

We have that

d^{(s)}(μ) = |D^{(s)}(μ)| = ∑_{{i≥1,ℓ(i)=s}} d(m_{i}(μ))

and

d(μ) = ∑_{s=1}^{r−1} d^{(s)}(μ) = |D(μ)| .

Consider nonzero residues s, t modulo r, let μ = (i^{m_{i}(μ)}) and define

T^{(st)}(μ) = {(i, j) | 1 ≤ i, 1 ≤ j ≤ m_{i}(μ), ℓ(i) = s, ℓ(j) = t}.

Glaisher [6] defined a bijection between the sets C and R of class regular and regular partitions of n. Glaisher’s map G is defined as follows. Suppose that μ = (i^{m_{i}(μ)}) ∈ C. Consider the r-adic expansion of each multiplicity m_{i}(μ) :

m_{i}(μ) = ∑_{j≥0} m_{ij}(μ)r^{j}

where for all relevant i, j we have m_{ij}(μ) ∈ {0, …, r − 1}. Then G(μ) = ρ where for all i, j, r ∤ i we have m_{ir^{j}}(ρ) = m_{ij}(μ).

We show

Proposition 7 If μ ∈ C then |T^{(st)}(μ)| = |T^{(st)}(G(μ))| + d^{(s)}(μ).

Proof. We establish a bijection δ^{(st)}(μ) between T^{(st)}(μ) and the disjoint union T^{(st)}(G(μ)) ∪ D^{(s)}(μ). If (i, j) ∈ T^{(st)}(μ) and (k(j), q(j)) = (a, q), we have two possibilities

(i) j is a defect number for m_{i}(μ). Then we map (i, j) onto (i, a, q) ∈ D^{(s)}(μ).

(ii) We have j = tr^{a} + h_{a+1}(m_{i}(μ)) where 1 ≤ t ≤ m_{ia}(μ). Then we map (i, j) onto (r^{a}i, t) ∈ T^{(st)}(G(μ)).

This establishes the desired bijection. □

Consider nonzero residues s, t modulo r, and define

T_{C}^{(st)} = {(μ, i, j)|μ ∈ C, (i, j) ∈ T^{(st)}(μ)}
T_{R}^{(st)} = {(ρ, i, j)|ρ ∈ R, (i, j) ∈ T^{(st)}(ρ)}
D^{(s)} = {(μ, i, a, q)|μ ∈ C, (i, a, q) ∈ D^{(s)}(μ)}.

Clearly the bijections δ^{(st)}(μ), μ ∈ C, above induce a bijection

δ^{(st)} : T_{C}^{(st)} ⟷ T_{R}^{(st)} ∪ D^{(s)}.

Putting the bijections δ^{(ts)}, t = 1, …, r −1 together we obtain a bijection

δ^{(s)} : ⋃_{t=1}^{r−1} T_{C}^{(ts)} ⟷ ⋃_{t=1}^{r−1} T_{R}^{(ts)} ∪ C,

where

C = ⋃_{t=1}^{r−1} D^{(t)}.

In [12, proof of Theorem 4], an involution ι was defined on the set

T_{C} = {(μ, i, j)|μ ∈ C, i, j ≥ 1, m_{i}(μ) ≥ j}.

From the definition of ι it follows that it maps the subset T_{C}^{(st)} of T_{C} into T_{C}^{(ts)}. Thus we conclude Lemma 8 For all s ∈ {1, …, r − 1} there is a bijection

ι^{(s)} : ⋃_{t=1}^{r−1} T_{C}^{(st)} ⟷ ⋃_{t=1}^{r−1} T_{C}^{(ts)} .

Composing the bijections ι^{(s)} and δ^{(s)} we see Proposition 9 For all s ∈ {1, …, r − 1} there is a bijection

κ^{(s)} : ⋃_{t=1}^{r−1} T_{C}^{(st)} ⟷ ⋃_{t=1}^{r−1} T_{R}^{(ts)} ∪ C.

Proof of Proposition 6. Just consider the cardinalities of the sets occurring in Proposition 9.

|⋃_{t=1}^{r−1} T_{C}^{(st)}| = ∑_{μ∈C} ∑_{{i|ℓ(i)=s}} m_{i}(μ) = α_{C}^{(s)}.

The latter equality holds because a class regular partition contains no parts divisible by r. Thus if m_{i}(μ) ≠ 0 then ℓ(i) = s if and only if i ≡ s(mod r).

|⋃_{t=1}^{r−1} T_{R}^{(ts)}| = ∑_{ρ∈R} |{i|m_{i}(ρ) ≥ s}| = β_{R}^{(s)}.

This is because parts in regular partitions have multiplicities < r. Finally

|C| = ∑_{t=1}^{r−1} d^{(t)} = ∑_{μ∈C} d(μ) = d_{C}. □

Remark. There is of course also a singular character table for S_{n}, which we denote X_{R′C′}. It is also a submatrix of the character table X of S_{n}. The subscript R′C′ indicates that the rows of X_{R′C′} are indexed by the set R′ of singular (i.e. nonregular) partitions of n, and the columns by the set C′ of class singular (i.e. non-class regular) partitions of n. For this we have

| det (X_{R′C′}) | = b_{C′}.  (4)

There are different ways of proving this. In [12] there is a proof based on Theorem 1 and a result in [9].

Another way of proving (4) is via an identity of Jacobi [5, p. 21]. Namely, suppose that A is an invertible n × n matrix, and write A and A^{−1} in the block form

A = [ \begin{matrix}B & C \\ D & E \end{matrix} ] , A^{−1} = [ \begin{matrix}B′ & C′ \\ D′ & E′ \end{matrix} ] ,

where B and B′ are k × k matrices. Then

det E′ = \frac{det B}{det A} .

By the orthogonality of characters we have

X^{−1} = X^{t}Δ(z_{μ}^{−1}),

where Δ(z_{μ}^{−1}) is the diagonal matrix with the z_{μ}^{−1}, μ ∈ P, on the diagonal. Equation (4) follows immediately from this observation and Theorem 1.

Remark. If we keep r fixed and let n vary, then the result of Proposition 6 may also be proved by calculating the generating functions for α_{C}^{(s)}, β_{R}^{(s)} and d_{C}. Indeed, if P (q) is the generating function for the number of partitions of n, then P_{r}(q) = \frac{P (q)}{P (q^{r})} is the generating function for the number of regular

partitions of n. We may then express the generating functions for α_{C}^{(s)}, β_{R}^{(s)} and d_{C} respectively by

A^{(s)}(q) = P_{r}(q) ∑_{i≥0} \frac{q^{ir+s}}{1 − q^{ir+s}}

B^{(s)}(q) = P_{r}(q) ∑_{j≥1} \frac{q^{js} − q^{jr}}{1 − q^{jr}}

D(q) = P_{r}(q) ∑_{j≥1} \frac{q^{jr}}{1 − q^{jr}} .

We omit the details. From this Proposition 6 may be deduced easily.

3 Smith normal forms of character tables related to S_{n}

For a partition λ of n, we denote by ξ^{λ} the permutation character of S_{n} obtained by inducing the trivial character of the Young subgroup S_{λ} up to S_{n}. First we explicitly describe the values of these permutation characters (this is included here as we have not been able to find a reference for it).

Proposition 10 Let λ, μ ∈ P, k = ℓ(λ), ℓ = ℓ(μ). Then the value ξ^{λ}(μ) of the permutation character ξ^{λ} on the conjugacy class of cycle type μ equals the number of ordered set partitions (B_{1}, …, B_{k}) of {1, …, ℓ} such that

λ_{j} = ∑_{i∈B_{j}} μ_{i} for j ∈ {1, …, k} .

Proof. Let σ_{μ} be a permutation of cycle type μ. Then (see [8]) ξ^{λ}(μ) is the number of λ-tabloids fixed by σ_{μ}. Now clearly, a λ-tabloid is fixed by σ_{μ} if and only if its rows are unions of complete cycles of σ_{μ}. Thus such a decomposition of rows corresponds to an ordered set partition (B_{1}, …, B_{k}) of the cycles of μ with the sum conditions in the statement of the Proposition. □

Remark. One may also use a symmetric function argument for computing the values R_{λμ} = ξ^{μ}(λ). The complete homogeneous symmetric function h_{λ} is the (Frobenius) characteristic of the character ξ^{λ} (see [14, Cor. 7.18.3]), so h_{λ} = ∑_{μ} z_{μ}^{−1}R_{λμ}p_{μ}. As the h_{λ} and m_{μ} are dual bases, as well as the p_{λ} and z_{μ}^{−1}p_{μ}, it then follows that p_{λ} = ∑_{μ} R_{λμ}m_{μ}. Using [14, Prop. 7.7.1] then also gives the formula in Proposition 10.

Corollary 11 Let λ, μ ∈ P. Then we have

(i) ξ^{λ}(μ) = 0 unless λ ≥ μ (dominance order).

(ii) ξ^{λ}(λ) = b_{λ} = ∏_{i} m_{i}(λ)!.

(iii) ξ^{λ}(λ) | ξ^{λ}(μ).

Proof. Using the remark above, parts (i) and (ii) follow immediately by [14, Cor. 7.7.2] (or one may also prove it directly using Proposition 10). For (iii), we use the combinatorial description given in Proposition 10. With notation as before, let (B_{1}, …, B_{k}) be an ordered partition of the set {1, …, ℓ} contributing to ξ^{λ}(μ), i.e., satisfying the sum conditions. Now any permutation of {1, …, k} which interchanges only parts of λ of equal size leads to a permutation of the entries of (B_{1}, …, B_{k}) such that the corresponding ordered partition still satisfies the sum conditions. Hence ξ^{λ}(μ) is divisible by ∏_{i} m_{i}(λ)! = b_{λ} and thus by ξ^{λ}(λ). □

We can now determine the Smith normal form for the regular character table of S_{n} in the case where r = p is prime.

For an integer matrix A we denote by S(A) its Smith normal form. If p is a prime, we write A_{p′} for the matrix obtained by taking only the p′-parts of the entries. For a set of integers M = {r_{1}, …, r_{m}} we denote by S(M) or S(r_{1}, …, r_{m}) the Smith normal form of the diagonal matrices with the entries r_{1}, …, r_{m} on the diagonal.

Theorem 12 Let p be a prime, and let X_{RC} be the p-regular character table of S_{n}. Then we have

S(X_{RC}) = S(b_{μ} | μ ∈ C)_{p′} .

Proof. Let Y = Y_{CC} = (ξ^{λ}(μ))_{λ,μ∈C} denote the part of the permutation character table of S_{n} with rows and columns indexed by the class p-regular partitions of n. Set X = X_{RC}. As the characters χ^{λ} with λ in the set R of p-regular partitions of n form a basic set for the characters on the p-regular conjugacy classes by [9], we have a decomposition matrix D = D_{CR} with integer entries such that

Y = D · X .

Now by Corollary 11 the permutation character table Y is (with respect to a suitable ordering) a lower triangular matrix with the b_{μ}, μ ∈ C, on the diagonal. Hence using [12, Theorem 4] and Theorem 1 we obtain

det(Y)_{p′} = (b_{C})_{p′} = a_{C} = | det(X)| .

Thus det(D) is a p-power, and hence det(D) and det(X) are coprime. This implies by [11, Theorem II.15]

S(Y) = S(D X) = S(D)S(X) .

Now using the divisibility property in Corollary 11 (iii) we can convert the triangular matrix Y by unimodular transformations to a diagonal matrix with the same entries b_{μ}, μ ∈ C, on the diagonal, and hence S(Y) = S(b_{μ} | μ ∈ C). As S(D) is a diagonal matrix with only p-power entries on the diagonal, this yields the assertion in the Theorem. □

Remark. Choosing p > n in Theorem 12 shows in particular that the Smith normal form of the whole character table X is the same as that of the diagonal matrix with diagonal entries b_{μ} = R_{μμ}, μ ∈ P. One may also use the language of symmetric functions to prove this result. Here, one uses that the matrix X is the transition matrix from the Schur functions to the power sums [14, Cor. 7.17.4]. Since the transition matrix from the monomial symmetric functions to the Schur functions is an integer matrix of determinant 1 (in fact, lower unitriangular with respect to a suitable ordering on partitions [14, Cor. 7.10.6]), the transition matrix R_{n} = (R_{λμ})_{λ,μ∈P} between the m_{λ}’s and p_{μ}’s has the same Smith normal form as X. Then we use the same arguments as before to deduce the Smith normal form of R_{n}.

Remark. We do not know at present how Theorem 12 should extend from the prime case to the case of general r. Some obvious guesses for rversions do not hold. The following weaker version might be true. Let π be the set of primes of r, and for a number m let m_{π′} denote its π′-part (the largest divisor of m coprime to r). Then

S(X_{RC})_{π′} = S(b_{μ} | μ ∈ C)_{π′} .

Using Theorem 12 above for p = 2 also allows the determination of the Smith normal form of the reduced spin character table of the double covers of the symmetric groups. For the background on spin characters of S_{n} we refer to [7] and [13].

We denote by D the set of partitions of n into distinct parts and by O the set of partitions of n into odd parts. Note that thus D is the set of 2-regular partitions of n and O is the set of class 2-regular partitions of n. For each λ ∈ D we have a spin character ⟨λ⟩ of S_{n}. If n − ℓ(λ) is odd, then there is an associate spin character ⟨λ⟩′ = sgn · ⟨λ⟩ of S_{n} and λ is said to be of negative type; the corresponding subset of D is denoted by D^{−}. The spin characters can have non-zero values only on the so-called doubling conjugacy classes of the double cover S̃_{n} of S_{n}; these are labelled by the partitions in O ∪ D^{−}. More precisely, for any such partition we have two conjugacy classes in S̃_{n}; one of these is chosen in accordance with [13], and we denote a corresponding representative by σ_{μ}. While the spin character values on the D^{−} classes are known explicitly (but they are in general not integers, and mostly not even real), for the values on the O-classes we only have a recursion formula (due to A. Morris) which is analogous to the Murnaghan-Nakayama formula, and which shows that these are integers. We then define the reduced spin character table as the integral square matrix

Z_{s} = (⟨λ⟩(σ_{μ}))_{{λ∈D \atop μ∈O}}

For any integer m ≥ 0, let s(m) be the number of summands in the 2-adic decomposition of m. For α = (1^{m_{1}}, 3^{m_{3}}, ⋯) ∈ O we define

k_{α} = ∑_{i odd}(m_{i} − s(m_{i})) .

Then we have

Theorem 13 The Smith normal form of the reduced spin character table Z_{s} of S̃_{n} is given by

S(Z_{s}) = S(2^{[k_{μ}/2]}, μ ∈ O) · S(b_{μ}, μ ∈ O)_{2′} .

Proof. Let Φ denote the Brauer character table of S̃_{n} at characteristic 2; this is equal to the Brauer character table of S_{n}. Then Z_{s} = D_{s} · Φ, where D_{s} is a “reduced” decomposition matrix at p = 2; the reduction corresponds to leaving out the associate spin characters ⟨λ⟩′ for λ ∈ D^{−}. The matrix D_{s} is then an integral square matrix. In [1], the Smith normal form of D_{s} was determined:

S(D_{s}) = S(2^{[k_{μ}/2]}, μ ∈ O) .

As this is a matrix of 2-power determinant and the determinant of the Brauer character table is coprime to 2, we have

S(Z_{s}) = S(D_{s}) · S(Φ) = S(2^{[k_{μ}/2]}, μ ∈ O) · S(Φ) .

Now the Brauer characters and the characters χ^{λ}, λ ∈ R = D, are both basic sets for the characters of S_{n} on 2-regular classes, hence S(Φ) = S(X_{RC}). By Theorem 12 (for p = 2) we thus obtain

S(Φ) = S(X_{RC}) = S(b_{μ} | μ ∈ O)_{2′} .

This proves the claim. □

Remark. Let us finally mention some open questions. We have determined the Smith normal form for the whole reduced spin character table.

It is natural to ask whether also a p-version (or even an r-version) of this holds, or at least, whether the determinant can be computed similarly as in the ordinary S_{n} case.

More precisely, for a prime p define

Z_{s,p} = (⟨λ⟩(σ_{μ}))_{{λ∈Dp \atop μ∈Op}}

where D_{p} and O_{p} denote the sets of class p-regular partitions in D and O, respectively. Some examples lead to the following conjecture:

S(Z_{s,p}) = S(2^{[k_{μ}/2]}, μ ∈ O_{p}) · S(b_{μ}, μ ∈ O_{p})_{2′} .

Concerning the determinant, one may ask whether there is an analogue of Theorem 4 in the spin case.

For S_{n} as well as its double cover one may also try to look for sectional versions or block versions for the results on regular character tables.

References

[1] C. Bessenrodt and J. B. Olsson, “Spin representations and powers of 2,” Algebras and Representation Theory 3 (2000), 289–300.

[2] C. Bessenrodt and J. B. Olsson, “A note on Cartan matrices for symmetric groups,” Arch. Math., 81 (2003), 497-504.

[3] J. Brundan and A. Kleshchev, “Cartan determinants and Shapovalov forms,” Math. Ann. 324 (2002), 431–449.

[4] S. Donkin, “Representations of Hecke algebras and characters of symmetric groups,” Studies in Memory of Issai Schur, Progress in Mathematics 210, pp. 158–170, Birkhäuser Boston, 2003.

[5] F. R. Gantmacher, The Theory of Matrices, vol. 1, Chelsea, New York, 1960.

[6] J. W. L. Glaisher, “A theorem in partitions,” Messenger of Math. 12 (1883), 158–170.

[7] P. Hoffman and J. F. Humphreys, Projective Representations of the Symmetric Groups, Oxford University Press, Oxford, 1992

[8] G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, New York, 1981.

[9] B. Külshammer, J. B. Olsson, and G. R. Robinson, “Generalized blocks for symmetric groups,” Invent. Math. 151 (2003), 513–552.

[10] I. G. Macdonald, Symmetric Functions and Hall Polynomials, second ed., Oxford University Press, Oxford, 1995.

[11] M. Newman, Integral Matrices, Academic Press, New York, 1972.

[12] J. B. Olsson, “Regular character tables of symmetric groups,” The Electronic Journal of Combinatorics 10 (2003), N3.

[13] I. Schur, “Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen,” J. reine ang. Math. 39 (1911) 155–250, (Gesammelte Abhandlungen 1, pp. 346–441, Springer-Verlag, Berlin/New York, 1973).

[14] R. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, New York/Cambridge, 1999.