Séminaire Lotharingien de Combinatoire 52 (2004), Article B52c

On character tables related to the alternating groups

Christine Bessenrodt

Institut für Mathematik, Universität Hannover D-30167 Hannover, Germany Email: bessen@math.uni-hannover.de

Jørn B. Olsson^{1}

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

Abstract

There is a simple formula for the absolute value of the determinant of the character table of the symmetric group S_{n}. It equals a_{P}, the product of all parts of all partitions of n (see [4, Corollary 6.5]). In this paper we calculate the absolute values of the determinants of certain submatrices of the character table X of the alternating group A_{n}, including that of X itself (Section 2). We also study explicitly the powers of 2 occurring in these determinants using generating functions (Section 3).

1. Preliminaries

We fix a positive integer n. We will use the same notation as in [2], which we recall here.

If μ = (μ_{1}, μ_{2}, …) is a partition of n we write μ ∈ P and then z_{μ} denotes 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} .

We consider the alternating group A_{n}. We let P^{+} denote the even partitions in P, O the partitions into odd parts, and D the partitions into distinct parts.

The conjugacy classes in A_{n} are of two types. The classes labelled by partitions μ ∈ P^{+} \ (O ∩ D) are the non-split classes, which contain all S_{n}-permutations of this type; we denote a representative by σ_{μ} and note that the corresponding centralizer is then of order z′_{μ} = z_{μ}/2. For the partitions μ ∈ D ∩O, the corresponding S_{n}-class splits into two conjugacy classes in A_{n}, for which we denote representatives by σ_{μ}^{+} and σ_{μ}^{−}; their centralizers are of order z′_{μ} = z_{μ}.

1Partially supported by The Danish National Research Council.

We briefly recall some information on the irreducible A_{n}-characters (see [5, sect. 2.5]).

Let μ be a partition of n. For μ ≠ μ̃, i.e., μ non-symmetric, [μ] ↓_{A_{n}}= [μ̃] ↓_{A_{n}} is irreducible. Let {μ} = {μ̃} denote this irreducible character of A_{n}. For μ = μ̃, i.e., μ symmetric, [μ] ↓_{A_{n}}= {μ}_{+} + {μ}_{−} is a sum of two distinct irreducible A_{n}-characters (which are conjugate in S_{n}). This gives all the irreducible complex characters of A_{n}, i.e.,

Irr(A_{n}) = {{μ}_{±} | μ ⊢ n, μ = μ̃} ∪ {{μ} | μ ⊢ n, μ ≠ μ̃} .

The characters {μ}_{±}, for symmetric μ, usually have non-rational values on the corresponding “critical” classes of cycle type h(μ) = (h_{1}, …, h_{l}), where h_{1}, …, h_{l} are the principal hook lengths in μ; note that h(μ) ∈ D ∩ O, so the corresponding S_{n}-class splits. Then we have [μ](σ_{h(μ)}) = (−1)^{\frac{n−l}{2}} =: ε_{μ} and

\{μ\}_{+}(σ_{h(μ)}^{±}) = \frac{1}{2} (ε_{μ} ± \sqrt{ε_{μ} ∏_{i=1}^{l} h_{i}})

\{μ\}_{−}(σ_{h(μ)}^{±}) = \frac{1}{2} (ε_{μ} ∓ \sqrt{ε_{μ} ∏_{i=1}^{l} h_{i}})

All other irreducible A_{n}-characters have the same value on these two classes.

For later use, we want to recall the Jacobi minor theorem (see [3, p. 21]). Let A = (a_{ij}) be an n × n matrix. Let M_{v} be a v-rowed minor of the determinant det A, corresponding to the rows i_{1}, …, i_{v} and the columns k_{1}, …, k_{v}. Then we take the (n − v)-rowed complementary minor for A by deleting all the rows and columns chosen for M_{v} before, and define the signed complementary minor M^{(v)} to M_{v} by multiplying this complementary minor by the sign ±1, depending on ∑_{j=1}^{v} i_{j} + ∑_{j=1}^{v} k_{j} being even or odd, respectively. (Note that for principal minors the sign is always +.)

Let A′ = (A_{ij}) be the n × n-matrix of cofactors A_{ij} for A, i.e., the adjoint matrix to A. Let M_{v} and M′_{v} be corresponding v-rowed minors of A and A′, respectively, then

M′_{v} = (det A)^{v−1}M^{(v)} .

2. Determinants of submatrices of the character table of A_{n}

We observe that by the Murnaghan-Nakayama formula we have for any symmetric partition μ and any ν ∈ D ∩ O:

{μ}_{±}(σ_{ν}^{±}) = 0 for all ν > h(μ) (in lexicographic order)

Hence, if we order the k (say) partitions in D ∩ O in decreasing lexicographic order, and the k symmetric partitions according to their principal hook lengths, then the corresponding 2k × 2k part of the character table of A_{n} is almost an upper triangular matrix, except that we have 2 × 2 blocks along the diagonal. We call this matrix X_{s}.

Knowing the entries of these diagonal blocks explicitly, we can easily compute their determinant and hence the (absolute value of the) determinant of this submatrix of the character table. A 2×2 block corresponding to the characters {μ}_{±} on the classes σ_{h(μ)}^{±} gives a contribution of absolute value

|ε_{μ}\sqrt{ε_{μ} ∏_{i} h_{i}}| = \sqrt{∏_{i} h_{i}} = \sqrt{a_{h(μ)}} ,

where h(μ) = (h_{1}, h_{2}, …). Hence the absolute value of the determinant of the whole submatrix is given by:

Proposition 2.1.

| det X_{s}| = ∏_{ν∈D∩O} \sqrt{a_{ν}} = \sqrt{a_{D∩O}} .

We can also easily determine the (absolute value of the) determinant for the whole character table X of A_{n}. By character orthogonality, we know that X̄^{t}X is a diagonal matrix with the centralizer orders as its diagonal entries. Set P^{(+)} = P^{+} \ (D ∩ O). Hence we have

\begin{aligned}| det X |^{2} & = & (∏_{μ∈P^{(+)}} \frac{z_{μ}}{2} ) (∏_{μ∈D∩O} z_{μ}^{2}) \\ & = & 2^{−|P^{(+)}|}z_{P^{+}}z_{D∩O} = 2^{−|P^{(+)}|}a_{P^{+}}b_{P^{+}}a_{D∩O} \end{aligned}

Now we have b_{P^{+}} = 2^{e^{+}}a_{P^{+}}, for some integer e^{+} ∈ ℤ. (This is not hard to prove by a combinatorial argument, see Lemma 3.3.)

Hence we obtain

Proposition 2.2.

| det X |^{2} = 2^{e^{+}−|P^{(+)}|}a_{P^{+}}^{2}a_{D∩O} .

In the next section we will see that e^{+} = e^{+}(n) ∈ ℕ, and that there is a nice generating function for the numbers e^{+}(n) (Proposition 3.4). In particular, an explicit formula for e^{+}(n) is given by

e^{+}(n) = ∑_{i=1}^{[n/2]} τ(i)p′(n − 2i) ,

where τ(i) is the number of divisors of i, and p′(j) = |D(j) ∩ O(j)|.

We are interested in determining the determinant of the integral part of the character table of A_{n} corresponding to the non-symmetric partitions and the non-split conjugacy classes; let us call this matrix X_{u} (with some ordering of rows and columns chosen). (Note that this is also a submatrix of the character table of S_{n}.) This part of the character table of A_{n} is complementary to the submatrix we have considered above, and we want to compute its determinant by employing Jacobi’s theorem.

Theorem 2.3. The determinant of the matrix X_{u} has absolute value

| det X_{u}| = 2^{(e^{+}−|P^{(+)}|)/2}a_{P^{(+)}} .

Proof. We assume that the rows of the character table X of A_{n} are labelled such that the rows corresponding to the symmetric partitions come first, and that the columns are labelled such that the v = |P^{(+)}| partitions in P^{(+)} come first. Let Δ be the diagonal matrix with the centralizer orders z′_{μ} as its diagonal entries, and let Δ^{(+)} be the diagonal submatrix corresponding to the partitions μ ∈ P^{(+)}.

As we have X̄^{t} · X = Δ , we know that the adjoint matrix to X is

X′ = (det X)Δ^{−1}X̄^{t} .

We now want to apply Jacobi’s minor theorem as it is stated in Section 1. We take the v-rowed minor M_{v} corresponding to the upper left square part in X, i.e., M_{v} = det X_{u}. The corresponding minor of X′ is then the determinant of

(det X)(Δ^{(+)})^{−1}X_{u}

(remember that X_{u} is integral). The signed complementary minor to M_{v} in X is then just det X_{s}. By Jacobi’s theorem we know that

(det X)^{v}( ∏_{μ∈P^{(+)}} z′_{μ})^{−1} det X_{u} = (det X)^{v−1} det X_{s} .

Hence

det X_{u} = (det X)^{−1}2^{−v}a_{P^{(+)}}b_{P^{(+)}} det X_{s} ,

and thus

\begin{aligned}| det X_{u}| & = & 2^{−(e^{+}−v)/2}(a_{P^{+}} \sqrt{a_{D∩O}})^{−1}2^{−v}a_{P^{(+)}}b_{P^{+}} \sqrt{a_{D∩O}} \\ & = & 2^{(e^{+}−v)/2}a_{P^{(+)}} \end{aligned}

where we have used the relation b_{P^{+}} = 2^{e^{+}}a_{P^{+}}. □

3. Powers of 2

We compute the generating functions for the powers of 2 occurring in the determinants of the previous section.

Let P (q), P^{+}(q), P^{−}(q) be the generating function for the number of partitions (resp. even/odd partitions) of n. The following is well-known:

Lemma 3.1. P^{+}(q) − P^{−}(q) = Δ(q), where

Δ(q) = ∏_{k≥0}(1 + q^{2k+1}) ( = \frac{P (q)P (q^{4})}{P (q^{2})^{2}})

is the generating function for the number of partitions of n into distinct odd parts.

Indeed, using that in P (q) = ∏_{k≥1} \frac{1}{1−q^{k}} the factor \frac{1}{1−q^{k}} accounts for the parts equal to k we see that

P^{+}(q) − P^{−}(q) = ∏_{k≥1} \frac{1}{1 + (−q)^{k}} .

Substituting q → −q in the Euler identity ∏_{k≥1}(1 + q^{k}) = ∏_{k≥0} \frac{1}{1−q^{2k+1}} and inverting we get

∏_{k≥1} \frac{1}{1 + (−q)^{k}} = ∏_{k≥0}(1 + q^{2k+1}) ,

proving the Lemma. □

We assume in the following always that δ = + or − is a sign.

Corollary 3.2. We have

P^{δ}(q) = \frac{P (q) + δΔ(q)}{2} .

We let P^{δ}(n) be the set of partitions of n with sign δ. Then define

a^{δ}(n) = a_{P^{δ}(n)} = ∏_{μ∈P^{δ}(n)} a_{μ}, b^{δ}(n) = b_{P^{δ}(n)} = ∏_{μ∈P^{δ}(n)} b_{μ} .

We factor each i ∈ ℕ as a product i = i_{2}i′, where i_{2} is a power of 2 and i′ is odd and consider two involutory bijections ι, ι′ on the set

T (n) = {(μ, d, k)|μ ∈ P(n), m_{d}(μ) ≥ k} .

Here

ι : (μ, d, k) ↦ (μ̂, k, d)

where μ̂ is obtained from μ by replacing k parts equal to d by d parts equal to k and leaving all other parts unchanged and

ι′ : (μ, d, k) ↦ (μ̃, d_{2}k′, k_{2}d′)

where μ̃ is obtained from μ by replacing k parts equal to d by k_{2}d′ parts equal to d_{2}k′ and leaving all other parts unchanged. Let

T_{d,k}^{δ}(n) = {μ ∈ P^{δ}(n)|m_{d}(μ) ≥ k} .

Then

(1)  |T_{d,k}^{δ}(n)| = p^{(−1)^{(d−1)k}δ}(n − dk)

Indeed removing k parts equal to d from a partition μ with sign δ gives you a partition with sign (−1)^{(d−1)k}δ and of cardinality |μ| − dk.

Note that this means that the partitions μ, μ̂ in the definition of ι have different signs if and only if (d − 1)k and d(k − 1) have different parities, ie. if and only if d and k have different parities. Moreover the partitions μ, μ̃ in the definition of ι′ have the same sign.

Thus ι induces a bijection between T_{d,k}^{δ}(n) and T_{k,d}^{δ}(n) if d, k have the same parity and between T_{d,k}^{δ}(n) and T_{k,d}^{−δ}(n) if d, k have different parities. Moreover the bijection ι′ shows that

(2)  a^{δ}(n)′ = b^{δ}(n)′,

and hence

Lemma 3.3. b^{δ}(n)/a^{δ}(n) = 2^{e^{δ}(n)} for some integer e^{δ}(n).

The power of 2 in a^{δ}(n) is

x^{δ}(n) = _{∏_{d,k} ; d even}d_{2}^{|T_{d,k}^{δ}(n)|}

and the power of 2 in b^{δ}(n) is

y^{δ}(n) = _{∏_{d,k} ; k even}k_{2}^{|T_{d,k}^{δ}(n)|}

Let x_{o}^{δ}(n), x_{e}^{δ}(n) be the product of the factors in x^{δ}(n), where k is odd/even and correspondingly y_{o}^{δ}(n), y_{e}^{δ}(n) be the product of the factors in y^{δ}(n), where d is odd/even. Using the map ι we see that

x_{e}^{δ}(n) = y_{e}^{δ}(n), x_{o}^{δ}(n) = y_{o}^{−δ}(n)

Thus the power of 2 in b^{δ}(n)/a^{δ}(n) is x_{o}^{−δ}(n)/x_{o}^{δ}(n). Suppose that x_{o}^{δ}(n) = 2^{f_{o}^{δ}(n)} and x_{e}^{δ}(n) = 2^{f_{e}^{δ}(n)}. Then e^{δ}(n) = f_{o}^{−δ}(n) − f_{o}^{δ}(n). We have (since ν_{2}(d) = 0, when d is odd)

f_{o}^{δ}(n) = _{∑_{d,k} ; k odd}ν_{2}(d)|T_{d,k}^{δ}(n)| = _{∑_{d,k} ; k odd}ν_{2}(d)p^{−δ}(n − dk) .

Let τ_{o}(n) the number of odd divisors of n. Note that τ_{o}(n)ν_{2}(n) equals the number τ_{e}(n) of even divisors of n. We then get (substituting dk = t in the above sum and noting that then ν_{2}(d) = ν_{2}(t) )

f_{o}^{δ}(n) = ∑_{t=1}^{n} τ_{o}(t)ν_{2}(t)p^{−δ}(n − t) = ∑_{t=1}^{n} τ_{e}(t)p^{−δ}(n − t) .

Let T (q) = ∑_{t≥1} \frac{q^{t}}{1−q^{t}} be the generating function for τ(n). Then T (q^{2}) is the generating function for the number τ_{e}(n) of even divisors of n. If F_{o}^{δ}(q) is the generating function for f_{o}^{δ}(n) we obtain

(3)  F_{o}^{δ}(q) = T (q^{2})P^{−δ}(q).

Using Lemmas 3.1 and 3.3 we deduce

Proposition 3.4. The generating function for e^{δ}(n) is

E^{δ}(n) = F_{o}^{−δ}(q) − F_{o}^{δ}(q) = δT (q^{2})Δ(q) .

Remark 3.5. This Proposition was also proved in [6] in a different way. Our approach was partially inspired by an unpublished note of John Graham. Note that the proposition shows that e^{+} = e^{+}(n) is always a positive integer.

Let us also consider F_{e}^{δ}(q). We have

f_{e}^{δ}(n) = ∑_{{d,k|k even}} ν_{2}(d)|T_{d,k}^{δ}(n)| = ∑_{{d,k|k even}} ν_{2}(d)p^{δ}(n − dk) .

We substitute dk = 2t in the above and obtain

f_{e}^{δ}(n) = ∑_{t≥1} τ^{∗}(t)p^{δ}(n − 2t) ,

where τ^{∗}(t) = ∑_{d|t} ν_{2}(d). We have

τ^{∗}(t) = \binom{ν_{2}(t) + 1}{2} ∏_{p odd}(ν_{p}(t) + 1) .

Thus if T^{∗}(q) is the generating function for τ^{∗}(t) then

F_{e}^{δ}(q) = T^{∗}(q^{2})P^{δ}(q) .

It is easily seen that

T^{∗}(q) = ∑_{j≥1} T (q^{2^{j}}) .

Proposition 3.6. The exponent of 2 in a^{δ}(n) has the generating function

F_{e}^{δ}(q) + F_{o}^{δ}(q) = T^{∗}(q^{2})P^{δ}(q) + T (q^{2})P^{−δ}(q) .

In Theorem 2.3 we have seen that | det X_{u}| = 2^{(e−|P^{(+)}|)/2}a_{P^{(+)}}. By Proposition 3.4, e = e^{+}(n) has generating function E^{+}(q) = Δ(q)T (q^{2}). Moreover |P^{(+)}(n)| has generating function P^{+}(q)−Δ(q) = P^{−}(q) (Lemma 3.1). Clearly, a_{P^{(+)}}(n) is divided by the same power of 2 as a^{+}(n), as the removed partitions have only odd parts. The generating function for the corresponding exponent is given by Proposition 3.6. Hence the exponent of 2 in det X_{u} has the generating function

G(q) = \frac{1}{2} (T (q^{2})Δ(q) − P^{−}(q)) + T^{∗}(q^{2})P^{+}(q) + T (q^{2})P^{−}(q),

and this then yields

Theorem 3.7. The exponent of 2 in det X_{u} has the generating function

G(q) = \frac{1}{2} (T (q^{2})P (q) − P^{−}(q)) + T^{∗}(q^{2})P^{+}(q) .

According to MAPLE the first values of the coefficients of G(q) are the following for n = 2, …, 14: 0 0 2 2 4 6 15 19 30 43 70 94 138.

Let us finally remark that the Propositions 3.4 and 3.6 also allow to compute the generating function for the exponent of 2 in | det(X)|, using Proposition 2.2.

References

[1] C. Bessenrodt, J. B. Olsson, A note on Cartan matrices for symmetric groups, Archiv d. Math. 81 (2003), 497–504.

[2] C. Bessenrodt, J. B. Olsson, R. P. Stanley, Properties of some character tables related to the symmetric groups, preprint 2003; to appear in: J. Algebraic Combinatorics.

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

[4] G. James, The representation theory of the symmetric groups, Lecture notes in mathematics 682, Springer–Verlag, 1978.

[5] G. James, A. Kerber, The Representation Theory of the Symmetric Group, Addison–Wesley, 1981.

[6] J. Müller, On a remarkable partition identity, J. Comb. Th. A 101 (2003), 271– 280.

[7] J. B. Olsson, Regular character tables of symmetric groups, The Electronic Journal of Combinatorics 10 (2003), Article #N3.