SQUARES IN THE CENTRE OF THE MODULAR GROUP ALGEBRA OF A FINITE SYMMETRIC GROUP

J. Murray* Mathematics Department, University College Dublin, Belfield Dublin 4, Ireland.

Let 𝒮(n) be a finite symmetric group of degree n and let F be a perfect field of characteristic p > 0. We use Z to denote the centre of the modular group algebra F 𝒮(n). If X is a subset of 𝒮(n) then X^{+} will denote its sum in F 𝒮(n). As is well known {K^{+} | K a class of 𝒮(n)} forms an F -basis for Z. We use Z_{p′} to denote the F -subspace of Z spanned by the p-regular class sums. The map Z → z^{p} is a semi-linear transformation on Z, with respect to the automorphism λ → λ^{p^{−1}} of F. Its image Z^{p} is an F -subalgebra of Z, and its kernel {z ∈ Z | z^{p} = 0} is an ideal of Z. The purpose of this paper is to prove the following:

Theorem 1. Let p = 2. Then Z^{2} = Z_{2′}. So z ∈ Z is a square in Z if and only if z is an F -linear combination of 2-regular class sums.

A p-block of 𝒮(n) is an indecomposable F -algebra, which is a direct summand of F 𝒮(n). Each p-block B of 𝒮(n) has an associated weight w and p-core α. So w is an integer between 0 and n/p, while α is a partition of n−wp which has no p-hooks. The numbers k(B) of irreducible characters, and l(B) of irreducible Brauer characters, in B, are both determined by w. The following is a blockwise version of Theorem 1:

Theorem 2. Let B be a 2-block of 𝒮(n), of weight w. Then Z^{2} ∩ Z(B) = Z_{2′} ∩ Z(B) and hence dim(Z^{2} ∩ Z(B)) = P (w).

Proof. Theorem 1 implies that Z^{2} ∩ Z(B) = Z_{2′} ∩ Z(B). But it is well known that dim(Z_{2′} ∩ Z(B)) = l(B) and that l(B) = P (w). The last statement follows.

Date: April 4, 2000.

1991 Mathematics Subject Classification. 20C20.

*the author was supported by an Enterprise Ireland research grant.

In fact, M. Enguehard has shown that w determines Z(B) up to isomorphism. See [E90]. So Theorem 2 is equivalent to Theorem 1.

It seems unlikely that one could find an explicit formula for the square root of a 2-regular class sum. However, we can at least show that:

Theorem 3. Each 2-regular class of 𝒮(n) occurs with odd multiplicity in the square of some involution class.

In fact, for each 2-regular class K of 𝒮(n), we can explicity describe a class I of involutions for which this theorem holds. Our method can be used to compute the square of any involution class sum of 𝒮(n).

We use the standard notation for subgroups, centralizers and normalizers. Let λ : F 𝒮(n) → F be the linear form λ(∑_{g∈𝒮} a_{g}g) = a_{1}, for ∑_{g∈𝒮} a_{g}g ∈

F 𝒮(n). The map (x, y) := λ(xy), for x, y ∈ F 𝒮(n), endows F 𝒮(n) with a nondegenerate symmetric bilinear form. Clearly (x, g) is the coefficient of g^{−1} in x, whenever x ∈ F 𝒮(n) and g ∈ 𝒮(n).

Fix an element g of 𝒮(g) and a p-defect group D of g. Let g_{p} be the p-part of g and let g_{p′} be the p-regular part of g. So g_{p} has p-power order and g_{p′} has p′-order, and g = g_{p}g_{p′} = g_{p′}g_{p}. We need the following:

Proposition 4. ⟨g_{p}⟩ is a direct factor of C_{𝒮(n)}(D).

In particular we may write

C_{𝒮(n)}(D) = ⟨g_{p}⟩ × N, (5)

where N is a normal subgroup of C_{𝒮(n)}(D).

Defering the proof of Proposition 4 for the moment, we proceed immediately to the proof of two corollaries. Corollary 7 will be needed in the proof of Theorem 1, while Corollary 6 may be of independent interest.

Corollary 6. Z_{p′} is a subalgebra of Z.

Proof. Let K and L be p-regular classes of 𝒮(n). It is enough to show that

(K^{+}L^{+}, g) = 0 if g_{p} ≠ 1.

So assume that g_{p} ≠ 1. By a standard argument we have

(K^{+}L^{+}, g) = ((C_{𝒮(n)}(D) ∩ K)^{+}(C_{𝒮(n)}(D) ∩ L)^{+}, g).

But every 2-regular element of C_{𝒮(n)}(D) is contained in the normal subgroup N given in (5). So

((C_{𝒮(n)}(D) ∩ K)^{+}(C_{𝒮(n)}(D) ∩ L)^{+}, g) = 0.

The corollary follows.

Let m be a nonnegative integer. Set Z(m) as the F -subspace of Z spanned by the class sums of elements of 𝒮(n) whose p-parts have order p^{m} or less. In particular Z(0) = Z_{p′}. The proof of the last corollary shows that Z(m) is a subalgebra of Z.

Let \sqrt[p]{g_{p′}} denote the unique p-regular element of 𝒮(n) such that (\sqrt[p]{g_{p′}})^{p} = g_{p′}.

Corollary 7. Z^{p} ⊆ Z_{p′}.

Proof. Let K be a conjugacy class of 𝒮(n). It is enough to show that

((K^{+})^{p}, g) = 0 if g_{p} ≠ 1.

Using a formula of Külshammer ((55) of [K91]) we have

((K^{+})^{p}, g) = |ϕ(K, g)| (modp),

where ϕ(K, g) := \{x ∈ 𝒮(n) | x^{p} = g_{p}, x\sqrt[p]{g_{p′}} ∈ K\}.

We claim that D acts on ϕ(K, g) by conjugation. For if x ∈ ϕ(K, g) and d ∈ D, then (x^{d})^{p} = (x^{p})^{d} = g_{p}^{d} = g_{p}, as D centralizes g_{p}. Also x^{d}\sqrt[p]{g_{p′}} = (x\sqrt[p]{g_{p′}})^{d} ∈ K, as D centralizes g_{p′}. This proves our claim.

Since D is a p-group, we get

|ϕ(K, g)| = |ϕ(K, g) ∩ C_{𝒮(n)}(D)| (modp).

But (5) implies that there is no x ∈ C_{𝒮(n)}(D) with x^{p} = g_{p} i.e. that ϕ(K, g) ∩ C_{𝒮(n)}(D) is empty. This completes the proof.

We highlight the following special case of this corollary.

Corollary 8. Let p = 2 and let I be a class of involutions in 𝒮(n). Then (I^{+})^{2} is a sum of 2-regular classes.

The conjugacy classes of 𝒮(n) are parametrized by the partitions of n. The p-regular classes correspond to the partitions of n into parts which are coprime to p. So if α is a partition of n into parts which are congruent to 1 modulo p, then the corresponding class of 𝒮(n) is p-regular. We let P (n) denote the number of partitions of n, and let P (n, p) denote the number of partitions of n into parts which are congruent to 1 modulo p.

Proposition 9. dim(Z^{p}) ≥ P (n, p).

Proof. This is a consequence of a construction of G. E. Murphy given in [M83]. Let K be an arbitrary integral domain. Murphy defines an element L_{u} of K𝒮(n) by

L_{u} := (1, u) + (2, u) + ⋯ + (u − 1, u),

where u be an integer between 2 and n, and (u, v) is a transposition. It is readily established that

L_{u} L_{v} = L_{v} L_{u},

for all u, v ∈ {2, …, n}. Let μ = (μ_{1}, μ_{2}, …, μ_{s}) be a partition of n and let g be an element of 𝒮(n) of cycle type μ. Set X^{μ} as the sum of all distinct products of the form

(L_{u_{1}})^{μ_{1}−1}(L_{u_{2}})^{μ_{2}−1} … (L_{u_{s}})^{μ_{s}−1},

where u_{1}, u_{2}, …, u_{s} runs over all sets of s elements from 2, 3, …, n. It turns out that X^{μ} lies in Z(K𝒮(n)). The main result of Theorem 1.9 of [M83] is that {X^{μ} | μ a partition of n} forms a K-basis for Z(K𝒮(n)). In fact, the coefficient of g in X^{μ} is 1, and if λ = (λ_{1}, λ_{2}, …, λ_{r}) is the cycle type of any element of 𝒮(n) which occurs in X^{μ}, then either s < r or μ ◁ λ, where ◁ is the dominance relation on partitions.

Let K = F. Choose μ so that

μ_{i} ≡ 1 (mod p), (i = 1, 2, …, s).

Since the L_{u} commute, each term in X^{μ} is a p-th power of the form

((L_{u_{1}})^{(μ_{1}−1)/p}(L_{u_{2}})^{(μ_{2}−1)/p} … (L_{u_{s}})^{(μ_{s}−1)/p})^{p},

Since F has characteristic p, and all the above terms commute with each other, we have X^{μ} = Y^{p}, where Y is a sum of terms of the form

(L_{u_{1}})^{(μ_{1}−1)/p}(L_{u_{2}})^{(μ_{2}−1)/p} … (L_{u_{s}})^{(μ_{s}−1)/p}.

But 1.9 of [M83] shows that Y lies in Z. So X^{μ} ∈ Z^{p}. The proposition now follows from the linear independence of the X^{μ}.

We can now prove our main theorem.

proof of Theorem 1. Clearly P (n, 2) equals the number of 2-regular classes of 𝒮(n). So Proposition 9 implies that dim(Z^{2}) ≥ dim(Z_{2′}). But Corollary 7 shows that Z^{2} ⊆ Z_{2′}. The theorem follows.

The proof of Theorem 3 is independent of much of the rest of the paper.

proof of Theorem 3. Assume that g is a 2-regular element of 𝒮(n). Let T be a class of involutions of 𝒮(n). Then by a standard argument we have

((T^{+})^{2}, g) = |{t ∈ T | g^{t} = g^{−1}}| (mod2).

Suppose that g has a_{i} commuting i-cycles in its cycle decomposition. Let t be an involution in 𝒮(n) which inverts g. Then t transposes 2b_{i}, and inverts c_{i}, commuting i-cycles in the cycle decomposition of g, where b_{i}, c_{i} are nonnegative integers and 2b_{i} + c_{i} = a_{i}. It follows that t has at least

∑_{i} b_{i}i + \frac{c_{i}(i − 1)}{2} = ∑ \frac{a_{i}(i − 1)}{2} + b_{i}

transpositions in its cycle decomposition. So the minimum number of transpositions needed to invert g is

N(g) := ∑ \frac{a_{i}(i − 1)}{2}.

Let S be the class of involutions in 𝒮(n) which are products of N(g) commuting transpositions, and let s be an involution in S which inverts g. Then s must invert each cycle in g, by the argument in the previous paragraph. In particular, s stabilizes each orbit of g on {1, 2, …, n}.

Suppose that X is a g-orbit. Then X is a union of s-orbits, and s fixes exactly one element x_{0} ∈ X. Clearly s transposes x_{0}g^{j} and x_{0}g^{1−j}. It follows that the action of s on X is determined by x_{0}. Conversely, given a set O of representatives for the orbits of g, we can find t ∈ S such that t inverts g and centralizes each member of O.

The previous paragraph shows that

|{t ∈ S | g^{t} = g^{−1}}| := ∏_{i} i^{a_{i}}.

The theorem now follows from the fact that ∏_{i} i^{a_{i}} is odd.

We prove Proposition 4 via a series of straightforward results. First we need some notation for subgroups of 𝒮(n). Our notation is taken from [R93, 1.6].

Let X and Y be finite sets. We use 𝒮(X) to denote the group of all permutations of X. By convention all permutations act on the right. Let H be a subgroup of 𝒮(X). If h ∈ H and y_{0} ∈ Y, we can define a permutation h(y_{0}) of X × Y via

(x, y)h(y_{0}) := \{ \begin{matrix}(xh, y), & if y = y_{0}, \\ (x, y), & if y ≠ y_{0}, \end{matrix} for all (x, y) ∈ X × Y.

The map h → h(y_{0}) gives an injection H ↪ 𝒮(X × Y), whose image we shall denote by H(y_{0}). We let H^{Y} denote the group generated by {H(y_{0}) | y_{0} ∈ Y }. So H^{Y} is the external direct product of |Y | copies of H.

Suppose that we have a collection of disjoint finite sets {X_{y} | y ∈ Y } and groups {H_{y} ≤ 𝒮(X_{y}) | y ∈ Y }, indexed by the elements of Y. Then ∏_{y∈Y} H_{y} will denote the group generated by {H_{y}(y) | y ∈ Y }. So ∏_{y∈Y} H_{y} is an embedding of the external direct product of the groups H_{y} in 𝒮(∪_{y}X_{y}).

Let K be a subgroup of 𝒮(Y). For k ∈ K, we can define a permutation k^{∗} of X × Y via

(x, y)k^{∗} := (x, yk), for all (x, y) ∈ X × Y.

The map k → k^{∗} gives an injection K ↪ 𝒮(X × Y), whose image we shall denote by Δ(K, X) (and Δ(K, n), if |X| = n). In particular, Δ(K, X) ≅ K.

The wreath product H≀K of H with K is the subgroup of 𝒮(X×Y) generated by H^{Y} and Δ(K, X). A quick calculation shows that H(y_{0})^{k^{∗}} = H(y_{0}k), for each y_{0} ∈ Y and k ∈ K. It follows that H^{Y} is a normal subgroup of H ≀ K. We call H^{Y} the base group of H ≀ K. Also H^{Y} ∩ Δ(K, X) = {1}. So H ≀ K is isomorphic to a semi-direct product of H^{Y} with K.

Let F(H) denote the set of elements of X which are fixed by every element of H, and let M(H) denote the complement of F(H) in X.

Lemma 10. C_{𝒮(X)}(H) = C_{𝒮(M(H))}(H) × 𝒮(F(H)). If F(H) = ϕ then

C_{𝒮(X×Y)}(H^{Y}) = C_{𝒮(X)}(H)^{Y} .

Proof. The first statement is a standard result.

Suppose that F(H) = ϕ. Then clearly C_{𝒮(X×Y)}(H^{Y}) ⊇ C_{𝒮(X)}(H)^{Y}. Let y_{0} ∈ Y and σ ∈ C_{𝒮(X×Y)}(H^{Y}). We have

F(H(y_{0})) = {(x, y) | x ∈ X, y ∈ Y \{y_{0}}}.

Since σ stabilizes F(H(y_{0})), it follows that (x, y_{0})σ = (xσ(y_{0}), y_{0}), for all x ∈ X, for some σ(y_{0}) ∈ 𝒮(X). Clearly σ(y_{0}) ∈ C_{𝒮(X)}(H). Thus σ = ∏_{y_{0}∈Y} σ(y_{0}) ∈ C_{𝒮(X)}(H)^{Y}. The lemma follows.

Lemma 11. Suppose that K acts transitively on Y. Then

C_{𝒮(X×Y)}(Δ(K, X)) = C_{𝒮(Y)}(K) ≀ 𝒮(X).

Proof. It is clear that C_{𝒮(X×Y)}(Δ(K, X)) ⊆ C_{𝒮(Y)}(K) ≀ 𝒮(X).

For each x ∈ X, the set x × Y := {(x, y) | y ∈ Y } is a Δ(K, X)-orbit on X × Y. Moreover, each Δ(K, X)-orbit equals x × Y, for some x ∈ X.

Let x ∈ X and σ ∈ C_{𝒮(X×Y)}(Δ(K, X)). The previous paragraph implies that (x × Y)σ = xσ_{1} × Y, for some σ_{1} ∈ 𝒮(X). So for y ∈ Y we have

(x, y)σ = (xσ_{1}, yσ_{x}(x)),

where σ_{x} ∈ 𝒮(Y) depends on x. An easy calculation shows that σ_{x} ∈ C_{𝒮(Y)}(K). So σ = σ_{1}^{∗} ∏_{x∈X} σ_{x}(x) lies in C_{𝒮(Y)}(K) ≀ 𝒮(X). The lemma follows.

Corollary 12. Suppose that H fixes no element of X and that K acts transitively on Y. Then C_{𝒮(X×Y)}(H ≀ K) = Δ(C_{𝒮(X)}(H), Y).

Proof. Using the definition of wreath product, and Lemmas 10 and 11, we have

\begin{aligned}C_{𝒮(X×Y)}(H ≀ K) & = C_{𝒮(X×Y)}(H^{Y}) ∩ C_{𝒮(X×Y)}(Δ(K, X)) \\ & = C_{𝒮(X)}(H)^{Y} ∩ C_{Y} (K) ≀ 𝒮(X). \end{aligned}

Thus

\begin{aligned}C_{𝒮(X×Y)}(H ≀ K) & = ⟨c(y_{0}) | c ∈ C_{𝒮(X)}(H), y_{0} ∈ Y ⟩ ∩ Δ(𝒮(X), Y) \\ & = Δ(C_{𝒮(X)}(H), Y). \end{aligned}

Recall that g is an element of 𝒮(n) and that D is a 2-defect group of g. We use these results to compute C_{𝒮(n)}(D). Suppose that g has a_{i} cycles of length i, for i = 1, 2, …, n. Let Z_{i} denote a cyclic subgroup of 𝒮(i) generated by an i-cycle.

Lemma 13. C_{𝒮(n)}(g) ≅ ∏_{i=1}^{n} Z_{i} ≀ 𝒮(a_{i}).

If n is an integer, write n_{p} for the p-part of n and n_{p′} for the p-regular part of n. So n = n_{p}n_{p′}, and n_{p} is a power of p, while n_{p′} is coprime to p. For i = 1, …, n let a_{i} = ∑_{j=0}^{∞} b_{ij}p^{j} be the base p-expansion of a_{i}, and let P (a_{i}) be a Sylow p-subgroup of 𝒮(a_{i}). It is well know that

P (a_{i}) ≅ ∏_{j} P (p^{j})^{b_{ij}}, (14)

where P (p^{j}) is a Sylow p-subgroup of 𝒮(p^{j}). Here we restrict j to those values for which b_{ij} ≠ 0. It is also known that P (p^{j}) is a transitive subgroup of 𝒮(p^{j}), and that the centre Z(P (p^{j})) of P (p^{j}) coincides with C_{𝒮(p^{j})}(P (p^{j})).

Lemma 15. D ≅ ∏_{i_{p}≠1} ∏_{j}(Δ(Z_{i_{p}}, i_{p′}) ≀ P (p^{j}))^{b_{ij}} × ∏_{i_{p}=1} Δ(P (a_{i}), i).

Proof. This follows from Lemma 13, (14), and the definition of the wreath product. Note that (Z_{i})_{p} = Δ(Z_{i_{p}}, i_{p′}), as subgroups of 𝒮(i). Proposition 16.

\begin{aligned}C_{𝒮(n)}(D) ≅ & & ∏_{i_{p}≠1} ∏_{j} Δ(Z_{i_{p}} ≀ 𝒮(i_{r}), p^{j})^{b_{ij}} \\ & × & ∏_{i_{p}=1} ∏_{j>0}(Z(P (p^{j})) ≀ 𝒮(i))^{b_{ij}} × 𝒮(∑_{i_{p}=1} ib_{i0}). \end{aligned}

Proof. Suppose that 1 ≤ i ≤ n and i_{p} ≠ 1. Then F(Δ(Z_{i_{p}}, i_{p′})) = ϕ, while Z_{i_{p}} is a transitive subgroup of 𝒮(i_{p}) and P (p^{j}) is a transitive subgroup of 𝒮(p^{j}). Thus

C_{𝒮(ip^{j})}(Δ(Z_{i_{p}}, i_{p′}) ≀ P (p^{j})) = Δ(C_{𝒮(i)}(Δ(Z_{i_{p}}, i_{p′})), p^{j}), by Corollary 12
= Δ(C_{𝒮(i_{p})}(Z_{i_{p}}) ≀ 𝒮(i_{p′}), p^{j}), by Lemma 11
= Δ(Z_{i_{p}} ≀ 𝒮(i_{p′}), p^{j}).

If i_{p} = 1, we have

Δ(P (a_{i}), i) = {1_{𝒮(ib_{i0})}} × ∏_{j>0} Δ(P (p^{j}), i)^{b_{ij}},

and F(Δ(P (p^{j}), i)) = ϕ, for j > 0. Also

C_{𝒮(ip^{j})}(Δ(P (p^{j}), i)) = C_{𝒮(p^{j})}(P (p^{j})) ≀ 𝒮(i), by Lemma 11
= Z(P (p^{j})) ≀ 𝒮(i).

The proposition now follows from repeated applications of Lemma 10.

proof of Proposition 4. Let H be a cyclic group which acts transitively and faithfully on a set X containing m = p^{a} elements, for some a ≥ 1. Let K be the full symmetric group on a set Y, where |Y | = r is coprime to p. Let h be a generator of H, and set d := ∏_{y∈Y} h(y). So Z(H ≀ K) = ⟨d⟩.

Every element of H ≀ K is of the form ∏_{y∈Y} h(y)^{α_{y}}k^{∗}, where k ∈ K, and 0 ≤ α_{y} ≤ m − 1, for each y ∈ Y. Define θ : H ≀ K → Z_{m} by

θ(∏_{y∈Y} h(y)^{α_{y}}k^{∗}) := ∑_{y∈Y} α_{y} (mod m).

Then θ is a group homomorphism, because h(y)k^{∗} = k^{∗}h(yk), for each k ∈ K, and y ∈ Y. For example, θ is the restriction of the sign homomorphism of 𝒮(X × Y) to H ≀ K when m = 2.

Now θ(d) = |Y | (mod m) ≢ 0 (mod m). So θ(⟨d⟩) = Z_{m}. Also d is central in H ≀ K. Thus

H ≀ K = ⟨d⟩ × Ker(θ). (17)

It follows from Proposition 16, and repeated applications of (17) that

C_{𝒮(n)}(D) ≅ ∏_{i_{p}≠1} ∏_{j} Z_{i_{p}}^{b_{ij}} × M

for some subgroup M of 𝒮(n). Also the projection of g_{p} onto each cyclic factor of

∏_{i_{p}≠1} ∏_{j} Z_{i_{p}}^{b_{ij}}

generates that factor. The proposition follows.

1. Acknowledgement

We wish to thank Gordan James for pointing out the significance of G. E. Murphy’s result.

References

[E90] M. Enguehard, Isométries parfaites entre blocs de groupes symétriques, S.M.F. Astérisque Vol. 181–182 (1990), p157–171.

[K91] B. Külshammer, Group-theoretical descriptions of ring-theoretical invariants of group algebras, Progress in Math. 95 (1991), 425-442.

[M83] G. E. Murphy, The Idempotents of the Symmetric Group and Nakayama’s Conjecture, J. Algebra 81 (1983), 258–265.

[R93] D. J. S. Robinson, A course in the theory of groups, Graduate Texts in Math. 80,

Springer-Verlag, New York, 1993.

  E-mail address: jcmurray@eircom.net