REGULAR CHARACTER TABLES OF SYMMETRIC GROUPS

JØRN B. OLSSON

It is a well-known fact that if X_{n} is the character table of the symmetric group S_{n}, then the absolute value of determinant of X_{n} equals a_{n}, which is defined as the product of all parts of all partitions of n. It also equals b_{n}, which is defined as the product of all factorials of all multiplicities of parts in partitions of n. Proofs of this may be found in [6], [5]. We sketch a proof below.

In this brief note we present generalizations of this to certain submatrices of X_{n} (called regular/singular character tables). We get such character tables for each choice of an integer ℓ ≥ 2. This is a perhaps slightly surprising consequence of results in [4]. The above result is obtained when we choose ℓ ≥ n.

If μ is a partition of n we write μ ⊢ n and then z_{μ} denotes the order of the centralizer of an element of (conjucagy) type μ in S_{n}. Suppose μ = (1^{m_{1}}, 2^{m_{2}}, ...), is written in exponential notation. Then we may factor z_{μ} = a_{μ}b_{μ}, where

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

We define

a_{n} = ∏_{μ⊢n} a_{μ}, b_{n} = ∏_{μ⊢n} b_{μ}

Proposition 1: We have that |det(X_{n})| = a_{n} = b_{n}.

Proof: (See also [6]) By column orthogonality for the irreducible characters of S_{n}, X_{n}^{t}X_{n} is a diagonal matrix with the integers z_{μ}, μ ⊢ n in the diagonal. It follows that in the above notation det(X_{n})^{2} = ∏_{μ⊢n} z_{μ} = a_{n}b_{n}. By [2], Corollary 6.5 we have |det(X_{n})| = a_{n}. The result follows.

Another proof of the fact that a_{n} = b_{n} for all n may be found in [3].

We choose an integer ℓ ≥ 2, which is fixed from now on. Several concepts below, like regular, singular, defect etc. refer to the integer ℓ.

A partition is called regular if no part is repeated ℓ or more times and is called class regular, if no part is divisible by ℓ. A partition which is not regular (class regular) is called singular (class singular). We let p(n) be the number of partitions of n. The number p^{∗}(n) of regular partitions of n equals the number of class regular partitions of n and then p′(n) = p(n) − p^{∗}(n) is the number of (class)singular partitions of n. The irreducible characters and the conjugacy classes of X_{n} are labelled canonically by the partitions of n. An irreducible character is called regular (singular), if the partition labelling it is regular (singular). A conjugacy class is called regular (singular), if the partition labelling it is class regular (class singular). The regular character table X_{n}^{reg} contains the values of regular characters on regular classes and the singular character table X_{n}^{sing} is defined analogously.

Authors address: Matematisk Afdeling, Universitetsparken 5, 2100 Copenhagen, Denmark.

Email: olsson@math.ku.dk.

Let

a_{n}^{reg} = ∏_{μ class regular} a_{μ}, b_{n}^{reg} = ∏_{μ class regular} b_{μ}

and define a_{n}^{sing} and b_{n}^{sing} correspondingly such that a_{n} and b_{n} are factored into a “regular” and a “singular” component, a_{n} = a_{n}^{reg} a_{n}^{sing}, b_{n} = b_{n}^{reg} b_{n}^{sing}.

Our main results are:

Theorem 2: The regular character table satisfies: |det(X_{n}^{reg})| = a_{n}^{reg}.

Theorem 3: The singular character table satisfies: |det(X_{n}^{sing})| = b_{n}^{sing}.

Remark: In the case where ℓ = p is a prime number, we have that the absolute value of the determinant of the Brauer character table of S_{n} in characteristic p is also a_{n}^{reg}.

When μ ⊢ n, say μ = (i^{m_{i}(μ)}) we define the defect of μ by

d_{μ} = ∑_{i,j≥1}⌊\frac{m_{i}(μ)}{ℓ^{j}}⌋,

where ⌊·⌋ means “integral part of.”

We start the proof of Theorems 2 and 3 with a key result which may be of independent interest. It generalizes the identity a_{n} = b_{n} above and is obtained by modifying an idea implicit in [6], see also [7], Exercise 26, p.48 and p.59. An unpublished note of of John Graham communicated to the author by Gordon James has been useful. The case where ℓ is a prime is implicit in [5], where proofs are based on modular representation theory.

Theorem 4: We have that b_{n}^{reg}/a_{n}^{reg} = ℓ^{c_{n}}, where

c_{n} = ∑_{μ class regular} d_{μ}.

Proof: Consider the set T of triples

T = {(μ, i, j)|μ class regular, i, j ≥ 1, m_{i}(μ) ≥ j}.

We claim that

a_{n}^{reg} = ∏_{(μ,i,j)∈T} i, b_{n}^{reg} = ∏_{(μ,i,j)∈T} j.

Indeed, for a fixed class regular μ and a fixed non-zero block i^{m_{i}(μ)} in μ, the elements (μ, i, 1), (μ, i, 2), ⋯, (μ, i, m_{i}(μ)) are precisely the ones in T starting with μ and i. These elements give a contribution i^{m_{i}(μ)} to a_{n}^{reg} and a contribution m_{i}(μ)! to b_{n}^{reg}.

We define an involutary bijection ι on T as follows. If (μ, i, j) ∈ T then ℓ does not divide i, since μ is regular and μ contains at least j parts equal to i. Write j = ℓ^{v}j′, where v is a non-negative integer and ℓ ∤ j′. We refer then to j′ as the ℓ′-part of j. Let μ_{(i,j)} be obtained from μ by replacing j parts equal to i in μ by ℓ^{v}i parts equal to j′. Then ι(μ, i, j) is defined as (μ_{(i,j)}, j′, ℓ^{v}i), an element of T . It is easily checked that ι^{2} is the identity.

This shows that

a_{n}^{reg} = ∏_{(μ,i,j)∈T} i = ∏_{(μ,i,j)∈T} j′,

where as above j′ is the ℓ′-part of j. Thus b_{n}^{reg}/a_{n}^{reg} = ℓ^{c}, where c is the sum of the exponents of the powers ℓ^{v} of ℓ, occuring as factors in the integers of the product ∏_{μ class regular,i≥1} m_{i}(μ)! If m is a positive integer, then there are ⌊m/ℓ⌋ numbers among 1, ⋯, m which are divisible by ℓ, ⌊m/ℓ^{2}⌋ numbers divisible by ℓ^{2}, etc., giving a total exponent ∑_{j≥1}⌊m/ℓ^{j}⌋ of ℓ in m!. Applying this fact to each m_{i}(μ), we get our result.

Let χ_{λ} denote the irreducible character of S_{n}, labelled by the partition λ ⊢ n, and χ_{λ}^{0} the restriction of χ_{λ} to the regular classes of S_{n}. In [4], Section 4, it was shown that there exist integers d_{λρ} such that for each irreducible character χ_{λ} we have

χ_{λ}^{0} = ∑_{ρ regular} d_{λρ}χ_{ρ}^{0}. (1)

It follows from (1) that for any λ the character

ψ_{λ} = χ_{λ} − ∑_{ρ regular} d_{λρ}χ_{ρ} (2)

vanishes on all regular classes.

Proof of Theorem 2: The matrix form of (1) above may be stated as

Y_{n} = D_{n}X_{n}^{reg},

where Y_{n} is the p(n)×p^{∗}(n)-submatrix of X_{n} containing the values of all irreducible characters on regular classes, and D_{n} = (d_{λρ}) is the “decomposition matrix”. Consider the corresponding “Cartan matrix” C_{n} = (D_{n})^{t}D_{n}. (For an explanation of the terms decomposition matrix and Cartan matrix we refer to [4].)

Column orthogonality shows that

(Y_{n})^{t}Y_{n} = (X_{n}^{reg})^{t}C_{n}X_{n}^{reg} = Δ(\frac{1}{z_{μ}}).

Taking determinants we see that

det(X_{n}^{reg})^{2}det(C_{n}) = ∏_{μ class regular} z_{μ} = a_{n}^{reg}b_{n}^{reg}  (3).

By Proposition 6.11 in [4] (see also [1], Theorem 3.3) we have that det(C_{n}) = ℓ^{c_{n}}. It follows then from Theorem 4 that

det(C_{n}) = b_{n}^{reg}/a_{n}^{reg}. (4)

From (3) and (4) we conclude |det(X_{n}^{reg})| = a_{n}^{reg}, which proves the theorem.

Proof of Theorem 3: We assume that the rows and columns of X_{n} are ordered such that the regular characters and classes are the first. Then the submatrix consisting of the intersection of the first p^{∗}(n) rows and the first p^{∗}(n) columns in X_{n} is exactly X_{n}^{reg}. In fact X_{n} has a block form

X_{n} = [ \begin{matrix}X_{n}^{reg} & A_{n} \\ B_{n} & X_{n}^{sing} \end{matrix} ]

We make some row operations on X_{n} to get a new matrix X̄_{n} as follows: Subtract for each singular partition λ′ and each regular partition ρ d_{λ′ρ} times the row labelled by ρ from the row labelled by λ′. Thus in X̄_{n} the row labelled by the singular partition λ′ contain the values of the character ψ_{λ′} on all conjugacy classes. Since ψ_{λ′} vanishes on regular classes X̄_{n} looks like this:

X̄_{n} = [ \begin{matrix}X_{n}^{reg} & A_{n} \\ 0 & Q_{n} \end{matrix} ]

for a suitable square p′(n)-matrix Q_{n}. We have then det(X_{n}) = det(X̄_{n}) = det(X_{n}^{reg}) det(Q_{n}), whence by Theorem 2

det(Q_{n}) = a_{n}^{sing}. (5)

We now have that if λ′, λ′′ are singular partitions, then since ψ_{λ′} vanishes on regular classes

< ψ_{λ′}, χ_{λ′′} >= ∑_{μ} \frac{1}{z_{μ}} ψ_{λ′}(x_{μ})χ_{λ′′}(x_{μ}) = ∑_{μ′ class singular} \frac{1}{z_{μ′}} ψ_{λ′}(x_{μ′})χ_{λ′′}(x_{μ′}).

Here x_{μ} is an element in the conjugacy class labelled by μ. On the other hand by (2) < ψ_{λ′}, χ_{λ′′} >= δ_{λ′λ′′}. Since Q_{n} and X_{n}^{sing} are both matrices with character values on singular conjucagy classes we see that

Q_{n}Δ(\frac{1}{z_{μ′}})(X_{n}^{sing})^{t} = E.

Here Δ is a diagonal matrix and E is a p′(n) unit matrix. We get for the determinants

det(X_{n}^{sing}) det(Q_{n}) = ∏_{μ′ class singular} z_{μ′} = a_{n}^{sing} b_{n}^{sing}  (6)

Now Theorem 3 follows from (5) and (6).

It should be remarked that Theorems 2 and 3 also holds, if we replace the irreducible characters χ_{λ} by Young characters η_{λ}.

Acknowledgements: The author thanks C. Bessenrodt and M. Schocker for discussions. This research was supported by the Danish Natural Science Foundation.

References

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