PROJECTIVE MODULES AND INVOLUTIONS

JOHN MURRAY

Abstract. Let G be a finite group, and let Ω := {t ∈ G | t^{2} = 1}. Then Ω is a G-set under conjugation. Let k be an algebraically closed field of characteristic 2. It is shown that each projective indecomposable summand of the G-permutation module kΩ is irreducible and self-dual, whence it belongs to a real 2-block of defect zero. This, together with the fact that each irreducible kG-module that belongs to a real 2-block of defect zero occurs with multiplicity 1 as a direct summand of kΩ, establishes a bijection between the projective components of kΩ and the real 2-blocks of G of defect zero.

Let G be a finite group, with identity element e, and let Ω := {t ∈ G | t^{2} = e}. Then Ω is a G-set under conjugation. In this note we describe the projective components of the permutation module kΩ, where k is an algebraically closed field of characteristic 2. By a projective component we mean an indecomposable direct summand of kΩ that is also a direct summand of a free kG-module. We show that all such components are irreducible, self-dual and occur with multiplicity 1.

This gives an alternative proof of Remark (2) on p. 254 of [5], and strengthens Corollaries 3 through 7 of that paper. In addition, we can give the following quick proof of Proposition 8 in [5]:

Corollary 1. Suppose that H is a strongly embedded subgroup of G. Then k_{H}↑^{G} ≅ k_{G}⊕[⊕_{i=1}^{s}P_{i}] where s ≥ 0 and the P_{i} are pairwise nonisomorphic self-dual projective irreducible kG-modules.

Proof. That H is strongly embedded means that |H| is even and |H ∩ H^{g}| is odd, for each g ∈ G\H. Let t ∈ H be an involution. Then clearly C_{G}(t) ≤ H. So k_{H}↑^{G} is isomorphic to a submodule of (k_{C_{G}(t)})↑^{G}. Mackey’s theorem implies that every component of k_{H}↑^{G}, other than k_{G}, is a projective kG-module. Being projective, these modules must be components of (k_{C_{G}(t)})↑^{G}. The result now follows from Theorem 8. □

Consider the wreath product G ≀ Σ of G with a cyclic group Σ of order 2. Here Σ is generated by an involution σ and G ≀ Σ is isomorphic to the semidirect product of the base group G × G by Σ. The conjugation action of σ on G × G is given by (g_{1}, g_{2})^{σ} = (g_{2}, g_{1}), for all g_{1}, g_{2} ∈ G. The elements of G ≀ Σ will be written (g_{1}, g_{2}), (g_{1}, g_{2}) σ or σ.

We shall exploit the fact that kG is a kG ≀ Σ-module. For, as is well-known, kG is an k(G × G)-module via: x · (g_{1}, g_{2}) := g_{1}^{−1}xg_{2}, for each x ∈ kG, and g_{1}, g_{2} ∈ G. The action of Σ on kG is induced by the permutation action of σ on the distinguished basis G of kG: g^{σ} := g^{−1}, for each g ∈ G. Clearly σ acts as an involutary k-algebra anti-automorphism of kG. It follows that the actions of G × G and Σ on kG are compatible with the group relations in G ≀ Σ.

Date: March 11, 2004.

1991 Mathematics Subject Classification. 20C20.

Key words and phrases. involutions, blocks of defect zero, Green correspondence, Burry-Carlson theorem.

By a block of kG, or a 2-block of G, we mean an indecomposable k-algebra direct summand of kG. Each block has associated to it a primitive idempotent in Z(kG), a Brauer equivalence class of characters of irreducible kG-modules and a Brauer equivalence class, modulo 2, of ordinary irreducible characters of G. A block has defect zero if it is a simple k-algebra, and is real if it contains the complex conjugates of its ordinary irreducible characters. Theorem 8 establishes a bijection between the real 2-blocks of G that have defect zero and the projective components of kΩ.

We could equally well work over a complete discrete valaution ring R of characteristic 0, whose field of fractions F is algebraically closed, and whose residue field R/J(R) is k. So we use O to indicate either of the commutative rings k or R.

All our modules are right-modules. We denote the trivial OG-module by O_{G}. If M is an OG-module, we use M↓_{H} to denote the restriction of M to H. If H is a subgroup of G and N is an OH-module, we use N↑^{G} to denote the induction of N to G. Whenever g ∈ G, we write \underline{g} for (g, g) ∈ G×G, and we set \underline{X} := \{\underline{x} | x ∈ X\}, for each X ⊂ G. Other notation and concepts can be found in a standard textbook on modular representation theory, such as [1] or [4].

If B is a block of OG, then so too is B^{o} = {x^{σ} | x ∈ B}. We call B a real block if B = B^{o}. Our first result describes the components of OG as OG ≀ Σ-module.

Lemma 2. There is an indecomposable decomposition of OG as OG ≀ Σ-module:

OG = B_{1} ⊕ … ⊕ B_{r} ⊕ (B_{r+1} + B_{r+1}^{o}) ⊕ … ⊕ (B_{r+s} + B_{r+s+1}^{o}).

Here B_{1}, …, B_{r} are the real 2-blocks and B_{r+1}, B_{r+1}^{o}, …, B_{r+s}, B_{r+s}^{o} are the nonreal 2-blocks of G.

Proof. This follows from the well-known indecomposable decomposition of OG, as an O(G × G)-module, into a direct sum of its blocks, and the fact that B_{i}^{σ} = B_{i} for i = 1, …, r, and B_{r+j}^{σ} = B_{r+j}^{o} for j = 1, …, s. □

An obvious but useful fact is that OG is a permutation module:

Lemma 3. The OG ≀ Σ-module OG is isomorphic to the permutation module (O_{\underline{G}×Σ})↑^{G≀Σ}.

Proof. The elements of G form a G≀Σ-invariant basis of OG. Moreover if g_{1}, g_{2} ∈ G, then g_{2} = g_{1} · (g_{1}, g_{2}). So G is a transitive G ≀ Σ-set. The stabilizer of e ∈ OG in G ≀ Σ is \underline{G} × Σ. The lemma follows from these facts. □

Let C be a conjugacy class of G. Set C^{o} := {c ∈ G | c^{−1} ∈ C}. Then C^{o} is also a conjugacy class of G, and C ∪ C^{o} can be regarded as an orbit of \underline{G} × Σ on the G ≀ Σ-set G. As such, the corresponding permutation module O(C ∪ C^{o}) is a O\underline{G} × Σ-direct summand of OG. If C = C^{o}, we call C a real class of G. In this case for each c ∈ C there exists x ∈ G such that c^{x} = c^{−1}. The point stabilizer of c in \underline{G} × Σ is \underline{C_{G}(c)}< \underline{x}σ >. So OC ≅ (O_{\underline{C_{G}(c)}<\underline{x}σ>})↑^{\underline{G}×Σ}. If C ≠ C^{o}, we call C a nonreal class of G. In this case the point stabilizer of c ∈ C ∪ C^{o} in \underline{G} × Σ is \underline{C_{G}(c)}. So O(C ∪ C^{o}) ≅ (O\underline{C_{G}(c)})↑^{\underline{G}×Σ}.

Suppose now that C_{1}, …, C_{t} are the real classes of G and that C_{t+1}, C_{t+1}^{o}, …, C_{t+u}, C_{t+u}^{o} are the nonreal classes. Then we have:

Lemma 4. There is a decomposition of OG as an O\underline{G} × Σ-permutation module:

OG = OC_{1} ⊕ … ⊕ OC_{t} ⊕ O(C_{t+1} ∪ C_{t+1}^{o}) ⊕ … ⊕ O(C_{t+u} ∪ C_{t+u+1}^{o}).

Proof. This follows from Lemma 3 and the discussion above. □

By a quasi-permutation module we mean a direct summand of a permutation module. Our next result is Lemma 9.7 of [1]. We include a proof for the convenience of the reader.

Lemma 5. Let M be an indecomposable quasi-permutation OG-module and suppose that H is a subgroup of G such that M↓_{H} is indecomposable. Then there is a vertex V of M such that V ∩ H is a vertex of M↓_{H}. If H is a normal subgroup of G, then this is true for all vertices of M.

Proof. Let U be a vertex of M. As O_{U} | M↓_{U} we have O_{U∩H} | (M↓_{H})↓_{U∩H}. But U ∩H is a vertex of O_{U∩H}. So Mackey’s Theorem implies that there exists a vertex W of M↓_{H} such that U ∩ H ≤ W.

As M↓_{H} is a component of the restriction of M to H, Mackey’s Theorem shows that there exists g ∈ G such that W ≤ U^{g} ∩ H. Now U^{g} is a vertex of M. So by the previous paragraph, and the uniqueness of vertices of M↓_{H} up to H-conjugacy, there exists h ∈ H such that U^{g} ∩ H ≤ W^{h}. Comparing cardinalities, we see that W = U^{g} ∩ H. So U^{g} ∩ H is a vertex of M↓_{H}.

Suppose that H is a normal subgroup of G. Then U ∩H ≤ W and W = U^{g}∩H = (U ∩ H)^{g} imply that U ∩ H = W. □

R. Brauer showed how to associate to each block of OG a G-conjugacy class of 2-subgroups, its so-called defect groups. It is known that a block has defect zero if and only if its defect groups are all trivial. J. A. Green showed how to associate to each indecomposable OG-module a G-conjugacy class of 2-subgroups, its so-called vertices. He also showed how to identify the defect groups of a block using its vertices as an indecomposable O(G × G)-module.

Corollary 6. Let B be a block of OG and let D be a defect group of B. If B is not real then \underline{D} is a vertex of B + B^{o}, as OG ≀ Σ-module. If B is real, then there exists x ∈ N_{G}(D), with x^{2} ∈ D, such that \underline{D} < \underline{x} σ > is a vertex of B, as OG ≀ Σ-module. In particular, Σ is a vertex of B + B^{o} if and only if B is a real 2-block of G that has defect zero.

Proof. J. A. Green showed in [2] that \underline{D} is a vertex of B, when B is regarded as an indecomposable O(G × G)-module. Suppose first that B is not real. Then B + B^{o} = (B↓_{G×G})↑^{G≀Σ}, for instance by Corollary 8.3 of [1]. It follows that B + B^{o} has vertex \underline{D}, as an indecomposable OG ≀ Σ-module.

Suppose then that B = B + B^{o} is real. Lemma 3 shows that B is \underline{G} × Σprojective. So we may choose a vertex V of B such that V ≤ \underline{G} × Σ. Moreover, B is a quasi-permutation OG ≀ Σ-module, and its restriction to the normal subgroup G × G is indecomposable. Lemma 5 then implies that V ∩ (G × G) = V ∩ \underline{G} is a vertex of B↓_{G×G}. So by Green’s result, we may choose D so that V ∩ \underline{G} = \underline{D}. Now G × G has index 2 in G ≀ Σ. So Green’s indecomposability theorem, and the fact that B↓_{G×G} is indecomposable, implies that V ⊈ (G × G). It follows that there exists x ∈ N_{G}(D), with x^{2} ∈ D, such that V = \underline{D} < \underline{x} σ >.

If B has defect zero, then D =<e>. So x^{2} = e. In this case, < \underline{x} σ > = Σ^{(e,x)} is G ≀ Σ-conjugate to Σ. So Σ is a vertex of B. Conversely, suppose that Σ is a vertex of B + B^{o}. The first paragraph shows that B is a real block of G. Moreover B has defect zero, as Σ ∩ \underline{G} =< e >. □

We quote the following result of Burry and Carlson [4, 4.4.6] on the Green correspondence:

Lemma 7. Let V ≤ H ≤ G be such that V is a p-group and N_{G}(V) ≤ H. Let f denote the Green correspondence with respect to (G, V, H). Suppose that M is an indecomposable OG-module such that M↓_{H} has a component N with vertex V. Then V is a vertex of M and N = f(M).

We can now prove our main result. Part (ii) is Remark (2) on p. 254 of [5], but our proof is independent of the proof given there.

Theorem 8. (i) Let t ∈ G, with t^{2} = e. Suppose that P is an indecomposable projective direct summand of (O_{C_{G}(t)})↑^{G}. Then P is irreducible and self-dual and occurs with multiplicity 1 as a component of (O_{C_{G}(t)})↑^{G}. In particular P belongs to a real 2-block of G that has defect zero. (ii) Suppose that M is a projective indecomposable OG-module that belongs to a real 2-block of G that has defect zero. Then there exists s ∈ G, with s^{2} = e, such that M is a component of (O_{C_{G}(s)})↑^{G}. Moreover, s is uniquely determined up to conjugacy in G.

Proof. If t = e then P = O_{G}. So P is irreducible and self-dual. The assumption that P is projective and the fact that dim_{O}(P) = 1 implies that |G| is odd. So all blocks of OG, in particular the one containing P, have defect zero.

Now suppose that t ≠ e. Let T be the conjugacy class of G that contains t. The permutation module OT is a direct summand of the restriction of OG to \underline{G} × Σ. Regard P as an O\underline{G}-module. Let I(P) be the inflation of this module to \underline{G} × Σ. Then I(P) is a component of OT. As Σ is contained in the kernel of I(P), and P is a projective OG-module, it follows that I(P) has vertex Σ as an indecomposable O\underline{G} × Σ-module.

By Lemma 2, and the Krull-Schmidt theorem, there exists a 2-block B of G such that I(P) is a component of the restriction (B + B^{o})↓_{\underline{G}×Σ}. An easy computation shows that N_{G≀Σ}(Σ) = \underline{G} × Σ. It then follows from Lemma 7 that (B + B^{o}) has vertex Σ and also that I(P) is the Green correspondent of (B + B^{o}) with respect to (G ≀ Σ, Σ, \underline{G} × Σ). We conclude from Corollary 6 that B is a real 2-block of G that has defect zero.

Let B̂ be the 2-block of G ≀ Σ that contains B. Then B̂ is real and has defect group Σ. Let Â be the Brauer correspondent of B̂. Then Â is a real 2-block of \underline{G} × Σ that has defect group Σ. Now Â = A ⊗ OΣ, where A is a real 2-block of O\underline{G} that has defect zero. In particular A has a unique indecomposable module, and this module is projective, irreducible and self-dual. Corollary 14.4 of [1] implies that I(P) belongs to Â. So P belongs to A. We conclude that P is irreducible and self-dual and belongs to a real 2-block of G that has defect zero.

Now B occurs with multiplicity 1 as a component of OG, and I(P) is the Green correspondent of B with respect to (G ≀ Σ, Σ, \underline{G} × Σ). So I(P) has multiplicity 1 as a component of the restriction of OG to \underline{G} × Σ. It follows that P occurs with multiplicity 1 as a component of (O_{C_{G}(t)})↑^{G}, and with multiplicity 0 as a component of (O_{C_{G}(r)})↑^{G}, for r ∈ G with r^{2} = e, but r not G-conjugate to t. This completes the proof of part (i).

Let R be a real 2-block of G that has defect zero. Then R has vertex Σ as indecomposable OG ≀ Σ-module. So its Green correspondent f(R), with respect to (G ≀ Σ, Σ, \underline{G} × Σ), is a component of the restriction of OG to \underline{G} × Σ that has vertex Σ. Lemma 4 and the Krull-Schmidt theorem imply that f(R) is isomorphic to a component of O(C ∪ C^{o}), for some conjugacy class C of G. Now Σ is a central subgroup of \underline{G} × Σ. So Σ must be a subgroup of the point stabilizer of C ∪ C^{o} in \underline{G} × Σ. It follows that s^{2} = e, for each s ∈ C. Let N denote the restriction of f(R) to \underline{G}, and consider N as an OG-module. We have just shown that N is a component of (O_{C_{G}(s)})↑^{G}. Argueing as before, we see that N is an indecomposable projective OG-module that belongs to a real 2-block of G that has defect zero.

The last paragraph establishes an injective map between the real 2-blocks of G that have defect zero and certain projective components of OΩ. As each block of defect zero contains a single irreducible OG-module, this map must be onto. It follows that the module M in the statement of the theorem is a component of some permutation module (O_{C_{G}(s)})↑^{G}, where s ∈ G and s^{2} = e. The fact that s is determined up to G-conjugacy now follows from the last statement of the proof of part (i). This completes the proof of part (ii). □

It is possible to simplify the above proof by showing that if B is a real 2-block of G that has defect zero, then its Green correspondent, with respect to (G≀Σ, Σ, \underline{G}×Σ) is M^{Fr}, where M^{Fr} is the Frobenius conjugate of the unique irreducible OG-module that belongs to B.

Corollary 9. Let Ω = {t ∈ G | t^{2} = e}. Then there is a bijection between the real 2-blocks of G that have defect zero and the projective components of OΩ.

Here is a sample application. It was suggested to me by G. R. Robinson.

Corollary 10. Let n ≥ 1 and let t be an involution in the symmetric group Σ_{n}. If n = m(m + 1)/2 is a triangular number, and t is a product of ⌊\frac{m^{2}+1}{4}⌋ commuting transpositions, then there is a single projective irreducible OΣ_{n}-module, and this module is the unique projective component of (O_{C_{Σn} (t)})↑^{Σ_{n}}. For all other values of n or nonconjugate involutions t, the modules (O_{C_{Σn} (t)})↑^{Σ_{n}} are projective free.

Proof. We give a proof of the following result in [3, Corollary 8.4]: Let G be a finite group, let B be a real 2-block of G of defect zero, and let χ be the unique irreducible character in B. Then there exists a 2-regular conjugacy class C of G such that C = C^{o}, |C_{G}(c)| is odd, for c ∈ C, and χ(c) is nonzero, modulo a prime ideal containing 2. Moroever, there exists an involution t ∈ G such that c^{t} = c^{−1}, and for this t we have < χ_{C_{G}(t)}, 1_{C_{G}(t)} >= 1. The existence of t was shown in [5]. The identification of t using the class C was first shown by R. Gow (in unpublished work).

Suppose that (O_{C_{Σn} (t)})↑^{Σ_{n}} has a projective component. Then Σ_{n} has a 2-block of defect zero, by Theorem 8. The 2-blocks of Σ_{n} are indexed by triangular partitions μ = [m, m − 1, …, 2, 1], where m ranges over those natural numbers for which n − m(m + 1)/2 is even. Moreover, the 2-block corresponding to μ has defect zero if and only if n = m(m + 1)/2. In particular, we can assume that n = m(m + 1)/2, for some m ≥ 1.

Let B be the unique 2-block of Σ_{n} that has defect zero, let χ be the unique irreducible character in B and let g ∈ Σ_{n} have cycle type λ = [2m − 1, 2m − 5, …]. Then |C_{Σ_{n}}(g)| is odd. As the parts of λ are the “diagonal hooklengths” of μ, the Murnaghan-Nakayama formula shows that χ(g) = 1. Now λ has ⌊(m − 1)/2⌋ nonzero parts. So g is inverted by an involution t that is a product of (n − ⌊(m − 1)/2⌋)/2 = ⌊\frac{m^{2}+1}{4}⌋ commuting transpositions. It follows from Theorem 8 and the previous paragraph that the unique irreducible projective B-module occurs with multiplicity 1 as a component of (O_{C_{Σn} (t)})↑^{Σ_{n}}. The last statement of the Corollary now follows from Theorem 8. □

References

[1] J. L. Alperin, “Local representation theory”, Camb. stud. adv. math. 11, 1986.

[2] J. A. Green, Blocks of modular representations, Math. Zeit. 79 (1962), 100–115.

[3] J. Murray, Extended defect groups and extended vertices, http://www.maths.may.ie/staff/jmurray/mypreprints.html (2003), 20 pages.

[4] H. Nagao, Y. Tsushima, “Representations of Finite Groups”, Academic Press, Inc. , 1989.

[5] G. R. Robinson, The Frobenius-Schur Indicator and Projective Modules, J. Algebra 126 (1989), 252–257.

Mathematics Department, National University of Ireland - Maynooth, Co. Kildare, Ireland.

  E-mail address: jmurray@maths.may.ie