Generalized blocks for symmetric groups

Jørn B. Olsson Copenhagen, Denmark

Talks in Durham (July 2002) and Beijing (September 2002)

Report on joint work with

Burkhard Külshammer and Geoffrey R. Robinson

Results from the representation theory of symmetric groups have analogues in other classes of groups and algebras. For instance there is an analogue of the Nakayama conjecture for Iwahori-Hecke algebras for the symmetric group S_{n} at an ℓ-th root of unity, ℓ ≥ 2 an integer. Donkin has presented a link between the representation theory of these algebras and an ℓ-analogue of the modular representation theory of the symmetric groups.

It thus seems natural to study “ℓ-blocks” of S_{n}. We descibe such a theory based primarily on the ordinary character theory of symmetric groups and on some very general ideas from the character theory of finite groups. We hope that this approach may provide additional insight, eg. concerning the invariant factors of Cartan matrices.

Blocks

G is a finite group, Irr(G) its set of complex irreducible characters and C is a union of conjugacy classes of G.

For complex-valued class functions α, β of G, we define

⟨α, β⟩_{C} := \frac{1}{|G|} _{y∈C}^{∑} α(y)\overline{β(y)}.

If ⟨α, β⟩_{C} = 0, we say that α and β are orthogonal across C.

If χ, μ ∈ Irr(G) are not orthogonal across C, then they are said to be directly C-linked.

A subset B of Irr(G) is said to be closed under C-linking, if whenever χ ∈ B and μ ∈ Irr(G) is directly C-linked to χ, then also μ ∈ B.

A C-block of G is a non-empty subset of Irr(G) which is minimal subject to being closed under C-linking. Irreducible characters of G in the same C-block are said to be C-linked.

Contributions

Γ(C) denotes the matrix of C-contributions, a square k(G) × k(G)-matrix with (i, j)-entry ⟨χ_{i}, χ_{j}⟩_{C}, where χ_{i}, χ_{j} ∈ Irr(G).

Also in this case this matrix may be expressed in different ways:

X(C) denotes k(G)×k(C)-submatrix of the complex character table X of G consisting only of the columns corresponding to classes in C. k(G): the number of conjugacy classes in G k(C): the number of conjugacy classes in C.

Then Δ(C) = \overline{X(C)}^{t}X(C) is a diagonal matrix whose i-th diagonal entry is |C_{G}(y_{i})|, where y_{i} is a representative for the i-th conjugacy class in C.

Let {ϕ_{r}|1 ≤ r ≤ k(C)} be any ℂ-basis for the space of complex-valued class functions of G which vanish identically on C′. Let Φ(C) be the (invertible) k(C) × k(C)-matrix with (i, j)-entry ϕ_{i}(y_{j}).

Let D(C) be the associated k(G) × k(C)-matrix of coefficients (analogous to the decomposition matrix of the usual block theory) such that X(C) = D(C)Φ(C). Then C(C) = \overline{D(C)}^{t}D(C) is analogous to the Cartan matrix of the usual block theory (however, below the term “Cartan matrix” is used only for the case when D is integral and the ϕ_{i}’s are generalized characters).

We then have the following expressions for the contribution matrix:

Γ(C) = D(C)C(C)^{−1} \overline{D(C)}^{t} =
X(C)[\overline{X(C)}^{t}X(C)]^{−1} \overline{X(C)}^{t} =
X(C)Δ(C)^{−1} \overline{X(C)}^{t}.

We note that Γ(C) is idempotent and that (since Γ(C)X(C) = X(C)) its rank (and trace) is k(C).

Invariant factors of the Cartan matrix

The union of conjugacy classes C is called closed if whenever x ∈ C, every generator of ⟨x⟩ is also in C. We will consider only closed unions of conjugacy classes.

For a complex-valued class function, θ, of G, let θ^{C} denote the class function of G which agrees with θ on C, and vanishes elsewhere.

Then R(C) denotes the ℤ-submodule of the space of complex class-functions of G generated by {χ^{C}|χ ∈ Irr(G)} and P(C) denotes the ℤ-submodule of R(C) consisting of generalized characters.

We have that R(C) and P(C) are both free Abelian of rank k(C)).

Let Cart(C) = R(C)/P(C), the Cartan group. This is a finite abelian group of exponent dividing |G| which can be generated by k(C) or fewer elements.

Note that R(C), P(C) and Cart(C) all have a direct sum decomposition corresponding to the C-blocks.

Let Cart(C, B) denote the summand corresponding to B, a union of C-blocks (and similarly for other notations).

Given a choice of ℤ-basis for P(C, B), say {θ_{i}|1 ≤ i ≤ s}, the Cartan matrix of B is the s × s matrix C(B) with (i, j)-entry ⟨θ_{i}, θ_{j}⟩_{C}. A different choice of ℤ-basis leads to a Cartan matrix C′(B) which satisfies C′(B) = A^{t}C(B)A for some unimodular integral matrix A, so the integral equivalence type of the associated quadratic form is well-defined (in particular, the invariant factors of the Cartan matrix are well-defined).

We have |Cart(C, B)| = |det(C(B))| and |Cart(C)| = |det(C(C))|.

We may descibe the relationship between the invariant factors of a Cartan matrix C(C) and of the corresponding matrix of contributions as follows.

Suppose that d_{1}|d_{2}| ⋯ |d_{s} are the invariant factors of C(C). Then M(C) := d_{s}Γ(C) is an integral matrix whose non-zero invariant factors are 1, \frac{d_{s}}{d_{s−1}}, …, \frac{d_{s}}{d_{2}}, \frac{d_{s}}{d_{1}}.

Perfect isometries

(Our definition appears to be slightly weaker than M. Broué’s definition. However, for our purposes the definition is strong enough.)

Suppose that C and D are closed unions of conjugacy classes of finite groups G and H respectively. Let B = B(C) and B′ = B′(D) be sets of irreducible characters of G, H which are closed under C-linking and D-linking respectively. Let Γ(C, B) and Γ(D, B′) be the contribution matrices associated to B and B′ respectively. We say that B and B′ are perfectly isometric if there is a diagonal matrix of signs S such that SΓ(C, B)S = Γ(D, B′).

Suppose then that B and B′ are perfectly isometric. Let B = {χ_{i}|1 ≤ i ≤ n} and B′ = {μ_{i}|1 ≤ i ≤ n}. The relationship between the contribution matrices just means that there are signs ε_{1}, …, ε_{n} such that ⟨χ_{i}, χ_{j}⟩_{C} = ⟨ε_{i}μ_{i}, ε_{j}μ_{j}⟩_{D} for all i, j.

We then have:

Under the above hypotheses, the Abelian groups R(C, B) and R(D, B′) are isomorphic via an isomorphism which restricts to an isomorphism between P(C, B) and P(D, B′). In particular, the Cartan groups Cart(C, B) and Cart(D, B′) are isomorphic. Moreoever with suitable choice of ℤ-bases, the Cartan matrices C(B) and C(B′) are equal.

We also have some general remarks on sections and blocks, which are omitted here.

On ℓ-blocks in S_{n} From now on G = S_{n}, ℓ ≥ 2 is an integer, π is the set π(ℓ) of primes dividing ℓ. ℓ-cycle element: All non-trivial cycles have length divisible by ℓ. ℓ-regular element: No cycle of length divisible by ℓ. ℓ-singular element At least one cycle of length divisible by ℓ. ℓ-element: An ℓ-cycle element with each nontrivial cycle of length dividing a power of ℓ. Moreover a π-regular element is an element whose order is not divisible by any prime in π.

Linked blocks and combinatorial blocks

We let S_{n}^{0} denote the set of ℓ-regular elements in S_{n}, a union of conjugacy classes. Then a (linked) ℓ-block is simply a C-block of S_{n}, where C = S_{n}^{0}.

The irreducible characters and the conjugacy classes of S_{n} are labelled canonically by the partitions of n. If λ is a partition of n then χ_{λ} denotes the irreducible character of S_{n}, labelled by λ.

We associate to λ its (ℓ-)core γ_{λ} and its (ℓ-)quotient β_{λ}. The ℓ-core is obtained from λ by removing all ℓ-hooks from λ. The number w_{λ} of ℓ-hooks to be removed from λ to go to the core is called the (ℓ-)weight of λ. The quotient β_{λ} is an ℓ-tuple of partitions, whose cardinalities add up to w_{λ}. It is known that β_{λ} and γ_{λ} determine λ uniquely.

A combinatorial ℓ-block of S_{n} consists by definition of all irreducible characters, whose partitions have a fixed ℓ-core.

When ℓ is a prime integer the “Nakayama conjecture” for S_{n} (which of course has been a theorem for many years) states that linked and combinatorial blocks are the same.

One of our main results is that the same is true for arbitrary ℓ. We thus in particular have yet another proof of the Nakayama conjecture.

It is fairly easy to show (using the Murnaghan-Nakayama recursion formula for character values) that if the irreducible characters χ_{λ} and χ_{λ′} are ℓ-linked, then λ and λ′ have the same ℓ-core. Thus a combinatorial block must be a union of linked blocks.

The other direction is difficult and is based on an Osima-type analysis of blocks.

On ℓ-projective characters and basic sets

As before S_{n}^{0} is the set of ℓ-regular elements of S_{n}. For any character (or class function) χ we write χ^{0} for χ^{S_{n}^{0}}.

Any ℤ-basis of R(S_{n}^{0}) is called an (ℓ-)basic set for S_{n}, and we refer to the elements of P(S_{n}^{0}) as (ℓ-)projective (generalized) characters of S_{n}.

A partition of n is called ℓ-regular if no part is repeated ℓ or more times. It is called ℓ-class regular if no part is divisible by ℓ.

We may then show (using ideas from section 6.3 of the James-Kerber book), that the χ_{μ}^{0}, where μ runs through the set of ℓ-regular partitions of n, form a basic set.

In fact the following is true:

For any irreducible character χ_{λ} of S_{n} we have

χ_{λ}^{0} = _{μ ℓ−regular}^{∑} d_{λμ}χ_{μ}^{0},

where d_{λμ}’s are integers satisfying the following properties:

(1) If d_{λμ} ≠ 0 then μ dominates λ^{R} and λ and μ have the same ℓ-core.

(2) If μ = λ^{R} then d_{λμ} = 1.

(3) For all ℓ-regular partitions μ the character ϕ_{μ} defined by

ϕ_{μ} = _{|λ|=n}^{∑} d_{λμ}χ_{λ}

is ℓ-projective.

There is an ℓ-regularization process associating to a partition λ its regularized partition λ^{R} obtained by moving all nodes on all ladders in λ to the top positions on the ladder. The partitions λ^{R} are ℓ-regular and a partition λ is ℓ-regular if and only if λ = λ^{R}. The longest ladder of λ is the ℓ-ladder with the highest number in λ.

Example. λ = (3, 2, 2, 2, 2, 1), ℓ = 4. Now λ is not 4-regular, since the part 2 is repeated 4 times.

0 1 2
3 0
2 3
1 2
0 1

3

To 4-regularize this we need just to move the node of residue 1 in position (5, 2) to position (2, 3). Thus λ^{R} = (3, 3, 2, 2, 1, 1).

Equivalence of ℓ-blocks

Consider a combinatorial ℓ-block of S_{n}.

Such a block B = B_{γ} consists of all characters χ_{λ} of S_{n} with a given ℓ-core γ. We refer to γ as the core γ(B) of B and the weight w = w(B) of B is the common ℓ-weight of the partitions labelling the characters in B. Thus if χ_{λ} ∈ B then

|λ| = w(B)ℓ + |γ(B)|.

We may prove that the Cartan matrices of ℓblocks of the same weight w have the same invariant factors. In fact they are perfectly isometric to the set of all irreducible characters of the wreath product ℤ_{ℓ} ≀S_{w}, with respect to a set of “regular conjugacy classes” (our name) considered by Osima.

This also allows us to prove that linked and combinatorial blocks are the same.

The sizes of the decomposition and Cartan matrices of B are given by the numbers k(B), (l(B)) of partitions (ℓ-regular partitions of n) with core γ(B). Thus

k(B) = k(ℓ, w(B)), l(B) = k(ℓ − 1, w(B)),

where generally k(r, w) is the number of rtuples of partitions whose cardinalities add up to w.

The rôle of u-numbers

We may write each element z ∈ S_{n} as z = x ∗ y, where x is an ℓ- cycle element, y is ℓ-regular and * signifies that x fixes the elements moved by y and vice versa. If the non-trivial parts of x are ℓr_{1}, ⋯, ℓr_{t} in decreasing order then ρ = (r_{1}, ⋯, r_{t}) is the type of z. In this case

χ_{λ}(x ∗ y) = _{|ν|=n−vℓ}^{∑} m_{λν}^{ρ}χ_{ν}(y)

where the MN-coefficients m_{λν}^{ρ} are integers.

By the above

χ_{ν}(y) = _{κ ℓ−regular}^{∑} d_{νκ}χ_{κ}(y).

Then with ρ as before and κ ℓ-regular we may define the u-numbers by

u_{λκ}^{ρ} = _{ν}^{∑} m_{λν}^{ρ}d_{νκ}.

If u_{λκ}^{ρ} ≠ 0 then λ and κ have the same ℓ-core.

It is possible to prove orthogonality realtions for these u-numbers, as it was done by Osima in the prime case. Their importance lie in the fact that they in a sense connect the characters in B with the characters in the wreath product. Suppose that ρ is a partition of v.

Consider the ”extreme” cases v = 0 and v = w, w = w(B). When v = 0, then the u-numbers are exactly the decomposition numbers for B,

When v > w the u-numbers are 0.

When v = w then u_{λγ}^{ρ} = m_{λγ}^{ρ} is closely related to the character value of an irreducible character of the wreath product S(ℓ, w) = ℤ_{ℓ} ≀ S_{w}.

The group S(ℓ, w) = ℤ_{ℓ} ≀ S_{w}. is a semi-direct product of a base subgroup ℤ_{ℓ}^{w} of order ℓ^{w} and a group S̃_{w} isomorphic to S_{w} operating on the base subgroup by place permutations. Two elements of S(ℓ, w) are called disjoint, if their S̃_{w}- factors are disjoint. Thus we get factorization of any element of S(ℓ, w) into “cycle factors”, according to the cycles of its S̃_{w}-factor. Such a cycle factor is characterized (up to conjugacy) by a pair (r, s), where r is the cycle length in S̃_{w} and s is a residue mod ℓ. It is conjugate to an element of S̃_{w} if and only if s = 0. The partitions in the label of a conjugacy class are obtained by collecting cycle factors with the same residue. In particular the parts of 0-th partition α_{0} of a conjugacy class label α describe the lengths r of those cycle factors which are conjugate to cycles inside S̃_{w}.

Thus the irreducible characters and the conjugacy classes of S(ℓ, w) are labelled canonically by the ℓ-quotients of w.

In this context Osima proved a generalization of the Murnaghan-Nakayama formula. When ℓ = 1, the result reduces to the usual MNformula for S_{w}. But in contrast to the S_{w}- case the MN-formula can only be applied to those conjugacy classes of S(ℓ, w), where the 0-th partition of the label is non-trivial. We call such a class singular of type ρ if the 0-th partition in its label is ρ. Otherwise the class is called regular.

Using the generalized MN-formula we may connect u-numbers and character values in S(ℓ, w).

Proposition Assume χ_{λ} ∈ B, |ρ| = w and that γ is the core of B. Let x be an ℓ-cycle element of type ρ. Then

σ_{λ}χ_{λ}(x) = ψ_{β_{λ}}(x̃)χ_{γ}(1),

where x̃ is in the S(ℓ, w)-conjugacy class labelled by (ρ, 0, ⋯, 0), (singular of type ρ). In particular

σ_{λ}u_{λγ}^{ρ} = ψ_{β_{λ}}(x̃).

Here σ_{λ} is a sign and ψ_{β_{λ}} is the irreducible character of S(ℓ, w) labelled by the ℓ-quotient of λ.

From this we may deduce:

Theorem: Let D(B) be the decomposition matrix of B and Z_{w}^{0} the submatrix of the character table of S(ℓ, w) consisting of the columns of regular conjugacy classes in the sense defined above. These are both k(B) × l(B)-matrices. Arrange the rows in these matrices such that the row in D(B) corresponding to χ_{λ} ∈ B and the row corresponding to ψ_{β_{λ}} in Z_{w}^{0} have the same number. Let σ(B) be the diagonal matrix with entries σ_{λ}, χ_{λ} ∈ B. Then there exists a non-singular complex matrix S(B) such that

σ(B)D(B) = Z_{w}^{0}S(B).

A perfect isometry

This theorem allows us to prove that characters with the same core are linked. Using the notation of this theorem we get by an easy calculation that

σ(B)D(B)C(B)^{−1}D(B)^{t}σ(B) =
Z_{w}^{0}[\overline{Z_{w}^{0}}^{t}Z_{w}^{0}]^{−1} \overline{Z_{w}^{0}}^{t}.

This yields a perfect isometry!

Indeed Γ(C, B) = D(B)C(B)^{−1}D(B)^{t} is the matrix of contributions for the set of irreducible characters in B with respect to C = S_{n}^{(0)} , (ℓregular classes in S_{n}.) Thus C-blocks are just linked ℓ-blocks.

Also Γ(D, B′) = Z_{w}^{0}[\overline{Z_{w}^{0}}^{t}Z_{w}^{0}]^{−1} \overline{Z_{w}^{0}}^{t} is the matrix of contributions for the set B′ of all irreducible characters of S(ℓ, w) with respect to D = S(ℓ, w)_{reg}, the union of the regular conjugacy classes (in the Osima sense) in S(ℓ, w). Thus

Theorem In the above notation B and B′ are perfectly isometric.

This implies also that if the latter matrix Γ(D, B′) is indecomposable in the Frobenius-Perron sense (ie, there is no relabelling of rows and columns so that the matrix has a proper decomposition into block form), then so is the former matrix Γ(C, B).

It may be shown (by a non-trivial calculation) that all characters of S(ℓ, w) are linked across regular elements. Therefore all irreducible characters of B are also linked across ℓ-regular elements of S_{n}, i.e. all irreducible characters in B are in the same linked block.

Thus

Theorem: The concepts of linked and combinatorial ℓ−blocks of S_{n} are identical.

Let us remark again that this theorem is the ℓanalogue of the Nakayama conjecture for symmetric groups, In the case where ℓ is a prime, the theorem appears to provide yet another new proof of the Nakayama conjecture.

Invariant factors

The starting point here is that the above results show:

Theorem: The invariant factors of the Cartan matrix C(B) of an ℓ-block B of S_{n} of weight w are equal to the invariant factors of the Cartan matrix of S(ℓ, w) with respect to regular classes. In fact the corresponding Cartan groups Cart(S_{n}^{0}, B) and Cart(S(ℓ, w)_{reg}) = Cart(ℓ, w) are isomorphic.

In particular, these groups certainly have the same exponent. The results of Donkin show that this exponent is a π-number, where π = π(ℓ) is the set of primes dividing ℓ. In fact we can show:

The exponent of Cart(ℓ, w) is ℓ^{w}w!_{π}.

Thus

The exponent of Cart(S_{n}^{0}) is ℓ^{⌊\frac{n}{ℓ}⌋}⌊\frac{n}{ℓ}⌋!_{π}. This is the also the largest invariant factor of an ℓ-Cartan matrix C_{n} of S_{n}.

The proof of these results reflect the potential usefulness of our approach. It may be easier to do calculations within S(ℓ, w) and to work with contributions (with respect to the set of regular elements) there.

A conjecture

In the prime case ℓ = p it is known that the invariant factors of the Cartan matrix are exactly the orders of the p-defect groups of p-regular conjugacy classes. This defect group is a direct product of the p-Sylow subgoups of wreath products ℤ_{m} ≀ S_{a}. Thus if a class regular partition is written exponentially (1^{a_{1}}, ⋯, m^{a_{m}}, ⋯), then each “block” m^{a_{m}} satisfying a_{m} ≥ p gives a contribution to the Cartan determinant, which is just the order of the p-Sylow subgroup of S_{a_{m}}.

In the general case there are obviously no defect groups of conjugacy classes or blocks. (In the case where ℓ is a power of the prime p, the group ℤ_{ℓ} ≀ P, P a p-Sylow subgroup of S_{w}, may in some sense be viewed as an “defect group” of an ℓ-block of weight w.).

The above suggests what the “ℓ-defect” of (1^{n}) should be. Numerous examples seem to indicate that in the composite case the contribution to the Cartan determinant of (m^{a}), where ℓ ∤ m, should not only depend on the multiplicity a, but also on the part m.

We use the notation λ ⊢ n (λ ⊢_{ℓ} n) to signify that λ is a (ℓ-class regular) partition of n. If m is a positive integer, we define ℓ_{m} = ℓ/(ℓ, m) and π_{m} as the set of primes dividing ℓ_{m}. If a is also a positive integer we set

r_{ℓ}(m, a) = ℓ_{m}^{⌊a/ℓ⌋} · ⌊a/ℓ⌋!_{π_{m}}.

Let the λ ⊢_{ℓ} n be written exponentially λ = (1^{a_{1}(λ)}, 2^{a_{2}(λ)}, ⋯). We define

r_{ℓ}(λ) = _{m}^{∏} r_{ℓ}(m, a_{m}(λ)). (1)

Conjecture: The abelian group Cart_{n} is a direct product of cyclic groups of order r_{ℓ}(λ), where λ runs through the set of ℓ-class regular partitions of n. In particular, the determinant of an ℓ-Cartan matrix of S_{n} is

det(C_{n}) = _{λ⊢_{ℓ}n}^{∏} r_{ℓ}(λ).

The invariant factors of the ℓ-Cartan matrices of S_{n} need not be powers of ℓ. When n < 2ℓ it is quite easy to see that only 1 and ℓ occur as invariant factors, in accordance with our conjecture. But when n ≥ 2ℓ, invariant factors occur which are not powers of ℓ. Here is an example. Exponents denote multiplicities of invariant factors.

Example: n = 18, ℓ = 6: The invariant factors of C_{n} are: 1296, 72^{9}, 18, 6^{55} , 3^{8}, 1^{223} . The r_{ℓ}(λ)’s for 6-regular partitions are: 1296, 72^{9}, 18, 6^{54} , 3^{9}, 2, 1^{222} . The determinant of C_{n} and the product of the r_{ℓ}(λ)’s equal 2^{87} · 3^{87} . The number 87 may be calculated by an explicit formula.

It may seem slightly surprising that the divisors ℓ_{m} of ℓ should occur, especially since the overall product of the r_{ℓ}(λ)’s is a power of ℓ.

Indeed in support of our conjecture it is possible to show that

det(C_{n}) = _{λ⊢_{ℓ}n}^{∏} r_{ℓ}(λ)

and that det(C_{n}) is a specifically given power of ℓ which may be described by a simple combinatorial formula.

Our proof is based on the work of Donkin and of Brundan and Kleshchev and of Bessenrodt and Olsson.

A conjecture of A. Mathas states that the determinant of the Cartan matrix of an Iwahori-Hecke algebra of S_{n} at an ℓ-th root of unity is a power of ℓ. Donkin showed that this Cartan matrix has the same determinant (and the same invariant factors) as the Cartan matrix C_{n} described above. Mathas’ conjecture was verified for ℓ-blocks of the Iwahori-Hecke algebras by Brundan and Kleshchev. This implies that det(C(B)) is a specific power of ℓ (depending on the weight w(B)) for each combinatorial ℓ-block B of S_{n}. This power may be described in two ways.

It can be shown that the determinant det(C(B)) calculated by Brundan and Kleshchev is the same as the one which was conjectured by Bessenrodt and Olsson before the work of Brundan-Kleshchev.

Therefore the conjecture of Bessenrodt-Olsson for det(C_{n}) also holds for arbitrary ℓ and based on this we may prove that the r_{ℓ}(λ)’s give the right determinant.

If w ≥ 0 we define

c_{ℓ}(w) = w ; ∑ ; j=0 l(j)k(ℓ − 2, w − j).

Then for all w ≥ 0 we have c_{ℓ}(w) = c_{ℓ}(w).

Define

c_{ℓ}^{∗}(n) = _{j≥1}^{∑} t(j)p^{∗}(n − jℓ),

where t(j) is the number of positive integer divisors in j and as usual p^{∗}(n) is the number of ℓ-(class)regular partitions of n. Then det(C_{n}) = ℓ^{c_{ℓ}^{∗}(n)}.

In support the above conjecture the following can be shown: r_{ℓ}(n) := ∏_{λ⊢_{ℓ}n} r_{ℓ}(λ) = ℓ^{c_{ℓ}^{∗}(n)}. Thus det(C_{n}) = ∏_{λ⊢_{ℓ}n} r_{ℓ}(λ)

On regular character tables

If μ is a partition of n 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}

Then it is well-known that |det(X_{n})| = a_{n} = b_{n}, where X_{n} is the character table of S_{n}.

A character of n is called regular if it is labelled by an (ℓ-)regular partiton (no part is repeated ℓ or more times). A conjugacy class is called regular if its conjugacy type is (ℓ-)class regular ie. no part is divisible by ℓ.

The regular character table X_{n}^{reg} contains the values of regular characters on regular classes.

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}.

Then

The regular character table satisfies:

|det(X_{n}^{reg})| = a_{n}^{reg}.

The singular character table satisfies:

|det(X_{n}^{sing})| = b_{n}^{sing}.