GENERIC JORDAN TYPE OF THE SYMMETRIC AND EXTERIOR POWERS

DAVID J. BENSON AND KAY JIN LIM

Abstract. We prove a result relating the stable generic Jordan types of the symmetric and exterior powers of the Heller translations of a module for a finite elementary abelian p-group. In the case of the trivial module, the stable generic Jordan types of the symmetric and exterior powers of its Heller translations are completely described.

1. Introduction

Fix an algebraically closed field k of positive characteristic p. Let E = ⟨g_{1}, …, g_{n}⟩ be an elementary abelian p-group of order p^{n} and M be a finitedimensional kE-module. The generic Jordan type of M is the Jordan type of

u_{α} = ∑_{i=1}^{n} α_{i}(g_{i} − 1)

on M for a generic element α = (α_{1}, …, α_{n}) ∈ 𝔸^{n}(k)\{0} (see [6]). The module M is of constant Jordan type if the Jordan type of u_{α} on M does not depend on α ∈ 𝔸^{n}(k)\{0} (see [5]).

In [5, Proposition 4.7], Friedlander, Pevtsova and Suslin proved that the generic Jordan type of the tensor product of kE-modules is the tensor product of their generic Jordan types. In [3, Corollary 4.3], Carlson, Friedlander and Pevtsova proved that the tensor product of two kE-modules of constant Jordan type is again a module of constant Jordan type. Motivated by these results, we asked, if the similar results hold, when tensor product is replaced by symmetric power S^{m} or exterior power Λ^{m}. The answers are negative. Suppose that m = p. We provide the regular kE-module kE as a counter-example of our queries.

Fix a positive integer m. Our main result relates the stable generic Jordan types of S^{m} and Λ^{m} of various Heller translations of a module for an elementary abelian p-group. As an application, we obtain a complete description of the stable generic Jordan types of S^{m}Ω^{i}k and Λ^{m}Ω^{i}k.

Throughout the article, we have omitted many known results which the readers can find in the standard texts [1, 4].

The second author is supported by Singapore Ministry of Education Academic Research Fund R-146-000-135-112.

2. Preliminaries

Let E be an elementary abelian p-group and M be a kE-module. Suppose that the generic Jordan type of M is [1]^{a_{1}}[2]^{a_{2}} ⋯ [p]^{a_{p}}. We call [1]^{a_{1}}[2]^{a_{2}} ⋯ [p− 1]^{a_{p−1}} the stable generic Jordan type of M. In the case M is generically free; i.e. a_{1} = ⋯ = a_{p−1} = 0, we write ∅ for its stable generic Jordan type.

Lemma 1. Let G be a finite group and N be the direct sum of r copies of the regular module kG. Let m be a positive integer such that 1 ≤ m ≤ r|G|. Suppose that lp^{d} = gcd(m, |G|) and p ∤ l. Then

(i)

V_{G}(Λ^{m}N) = ⋃_{\begin{matrix}H≤G, \\ |H| | lp^{d} \end{matrix}} Res_{G,H}^{∗} V_{H}(k) = V_{G}(S^{m}N).

In particular, the cohomological varieties V_{G}(Λ^{m}N), V_{G}(S^{m}N) have dimensions not more than d.

Suppose further that G is an elementary abelian p-group. Then

(ii) the cohomological varieties V_{G}(Λ^{m}N), V_{G}(S^{m}N) have dimensions

exactly d, and (iii) in the case of m = s|G|, the stable generic Jordan types of Λ^{m}N, S^{m}N

are [1]^{\binom{r}{s}}, [1]^{r^{s}} respectively.

Proof. (i) For each 1 ≤ j ≤ r, let N_{j} = kG. The module N = N_{1} ⊕ ⋯ ⊕ N_{r} has a canonical basis {g_{1,1}, …, g_{|G|,1}, …, g_{1,r}, …, g_{|G|,r}} where g_{i,j} ∈ G∩N_{j}. Then Λ^{m}N has a monomial basis Ω given by

{g_{i_{1},j_{1}} ∧ ⋯ ∧ g_{i_{m},j_{m}} | j_{1} ≤ ⋯ ≤ j_{m}, i_{1} ≤ ⋯ ≤ i_{m}, g_{i_{a},j_{a}} ≠ g_{i_{b},j_{b}} for a ≠ b}.

For each x = g_{i_{1},j_{1}} ∧ ⋯ ∧ g_{i_{m},j_{m}} ∈ Ω, let H_{x} = {g ∈ G | gx = ±x}. Then Λ^{m}N ≅ ⨁_{x∈Ω/G} Ind_{H_{x}}^{G} k_{x} for some one-dimensional kH_{x}-module k_{x}. Since H_{x} acts fixed-point freely on the set {g_{i_{1},j_{1}}, …, g_{i_{m},j_{m}}}, we have |H_{x}| | m; namely, |H_{x}| | lp^{d}. Conversely, given a subgroup H = {h_{1}, …, h_{t}} of G with t | lp^{d}, we let m = tl′ + s|G| where 1 ≤ tl′ ≤ |G|, for some non-negative integers s, l′. Since l′ ≤ |G|/t, we may choose {f_{1}, …, f_{l′}} ⊆ H\G and let

x = g_{1,1}∧⋯∧g_{|G|,1}∧⋯∧g_{1,s}∧⋯∧g_{|G|,s}∧h_{1}f_{1}∧⋯∧h_{t}f_{1}∧⋯∧h_{1}f_{l′}∧⋯∧h_{t}f_{l′}.

Clearly, x ∈ Ω and H ⊆ H_{x}. We now apply [4, Propositions 8.1.4, 8.2.4] to get the description of the cohomological variety V_{G}(Λ^{m}N).

By using the permutation basis

Ω′ = {g_{i_{1},j_{1}} ⋯ g_{i_{m},j_{m}} | j_{1} ≤ ⋯ ≤ j_{m} and i_{1} ≤ ⋯ ≤ i_{m}}

for S^{m}N, and arguing similarly, we get the other desired result. (ii) In this case, G has a subgroup E of order p^{d} and dim V_{E}(k) = d. (iii) Notice that Ind_{H_{x}}^{G} k_{x} is not generically free if and only if H_{x} = G. In this case, its generic Jordan type is [1] and there are precisely \binom{r}{s} of such x ∈ Ω fixed by G up to a sign. Similarly, there are precisely r^{s} of such x ∈ Ω′ fixed by G. □

Example 1. Suppose that E is elementary abelian of order p^{2}. The regular module kE is a module of constant Jordan type. On the other hand, by Lemma 1 (i), Λ^{p}kE has stable generic Jordan type non-∅ on the points of Res_{E,E′}^{∗} V_{E′}(k)\{0} for some subgroup E′ of order p of E, and ∅ otherwise. Thus Λ^{p}kE is not a module of constant Jordan type despite the fact that kE is. Similarly, we obtain that S^{p}kE is not a module of constant Jordan type.

Example 2. Again, let E be an elementary abelian p-group of order p^{2}. As we have seen in Example 1, Λ^{p}kE and S^{p}kE are generically free; namely their stable generic Jordan types are ∅. On the other hand, generically kE has Jordan type [p]^{p}. Taking the pth exterior power (respectively, symmetric power) of the generic Jordan type [p]^{p}, by Lemma 1 (iii), we obtain the stable generic Jordan type

[1]^{\binom{p}{1}} = [1]^{p} (respectively, [1]^{p^{1}} = [1]^{p}).

Thus, in general, generic Jordan type does not ‘commute’ with exterior power or symmetric power.

The remainder of this section is devoted to prove Lemma 3 which will be the key idea in the proof of Theorem 5.

Let C be a chain complex of kG-modules ⋯ → C_{i} ⟶^{d_{i}} C_{i−1} → · · ·. For a positive integer m, let X be the total complex of the m-fold tensor product of C over k; namely, the nth component X_{n} is the direct sum of C_{a_{1}} ⊗⋯⊗C_{a_{m}} where a_{1} + ⋯ + a_{m} = n, and the differential ∂_{n} : X_{n} → X_{n−1} is given by

∂_{n}(c_{1} ⊗ ⋯ ⊗ c_{m}) = ∑_{i=1}^{m}(−1)^{a_{1}+···+a_{i−1}}c_{1} ⊗ ⋯ ⊗ c_{i−1} ⊗ d_{a_{i}}c_{i} ⊗ c_{i+1} ⊗ ⋯ ⊗ c_{m}

where c_{i} ∈ C_{a_{i}}. Clearly ∂_{n−1} ∘ ∂_{n} = 0 and hence X is a chain complex of kG-modules. For each n, we have also a right 𝔖_{m}-action on X_{n} given by

(c_{1} ⊗ ⋯ ⊗ c_{m})(i, i + 1) = (−1)^{a_{i}a_{i+1}}c_{1} ⊗ ⋯ ⊗ c_{i+1} ⊗ c_{i} ⊗ ⋯ ⊗ c_{m}

for any 1 ≤ i ≤ m − 1. Direct calculation shows that the 𝔖_{m}-action commutes with ∂_{n} for any n ∈ ℤ and hence X is a chain complex of kG-k𝔖_{m}- bimodules.

Let 0 → M_{1} ⟶^{ι} M_{2} ⟶^{π} M_{3} → 0 be a short exact sequence of kG-modules. Specialize the above discussion to the chain complex C where

C_{i} = \{ \begin{matrix}M_{2} & i = 0, \\ M_{3} & i = −1, \\ 0 & otherwise; \end{matrix}

and d_{0} = π : M_{2} → M_{3}, we get the following lemma.

Lemma 2. Let 0 → M_{1} ⟶^{ι} M_{2} ⟶^{π} M_{3} → 0 be a short exact sequence of kG-modules. For every positive integer m, we have an exact sequence X^{m} of kG-k𝔖_{m}-bimodules

0 → X_{1}^{m} ⟶^{∂_{1}} X_{0}^{m} ⟶^{∂_{0}} X_{−1}^{m} ⟶^{∂_{−1}} ⋯ ⟶^{∂_{−m+1}} X_{−m}^{m} → 0,

where X_{1}^{m} = T^{m}(M_{1}) is the m-fold tensor product of the module M_{1}, ∂_{1} = ι^{⊗m} and, for each 0 ≤ j ≤ m, X_{−j}^{m} is the direct sum of the tensor products of exactly m − j and j copies of M_{2} and M_{3} respectively.

Proof. Let 0 ≤ j ≤ m. We want to prove that ker ∂_{−j} ⊆ im ∂_{−j+1}. It is done by induction on m. For 1 ≤ j ≤ m, we have X_{−j}^{m} = X_{−j}^{m−1} ⊗_{k} M_{2} ⊕X_{−j+1}^{m−1} ⊗_{k} M_{3}. Let {m_{1}, …, m_{r+s}} be a basis of M_{2} extending a basis {m_{1}, …, m_{r}} of M_{1}, and Ω, Ω′ be bases of X_{−j}^{m−1}, X_{−j+1}^{m−1} respectively. Suppose that

0 =∂_{−j} ( ∑_{x∈Ω, 1≤i≤r+s} a_{x,i}(x ⊗ m_{i}) + ∑_{x′∈Ω′, r+1≤i≤r+s} b_{x′,i}(x′ ⊗ π(m_{i})))

= ∑_{x∈Ω, 1≤i≤r+s} a_{x,i}(∂_{−j}x ⊗ m_{i}) + (−1)^{j} ∑_{x∈Ω, r+1≤i≤r+s} a_{x,i}(x ⊗ π(m_{i}))+
∑_{x′∈Ω′, r+1≤i≤r+s} b_{x′,i}(∂_{−j+1}x′ ⊗ π(m_{i})).  (1)

This shows that, for each 1 ≤ i ≤ r + s, ∑_{x∈Ω} a_{x,i}∂_{−j}x = 0 and hence, by induction, there exists y_{i} ∈ X_{−j+1}^{m−1} such that ∂_{−j+1}y_{i} = ∑_{x∈Ω} a_{x,i}x. Substitute

∂_{−j+1} ( ∑_{1≤i≤r+s} y_{i} ⊗ m_{i}) = ∑_{x∈Ω, 1≤i≤r+s} a_{x,i}(x⊗m_{i})+(−1)^{j−1} ∑_{i≥r+1} y_{i}⊗π(m_{i})

into Equation (1), we are reduced to the case of a_{x,i} = 0 for all x ∈ Ω. Similar argument shows that ∂_{−j+2}y′_{i} = ∑_{x′∈Ω′} b_{x′,i}x′ for some y′_{i} ∈ X_{−j+2}^{m−1} and hence we obtain that im ∂_{−j+1} = ker ∂_{−j}. The exactness at X_{0}^{m} can be proved similarly. □

We continue our discussion on the exact sequence X^{m} in Lemma 2. For each −1 ≤ j ≤ m, consider the tensor products X_{−j}^{m} ⊗_{k𝔖_{m}} k and X_{−j}^{m} ⊗_{k𝔖_{m}} sgn where sgn denotes the sign representation of 𝔖_{m}. Since the components M_{2}, M_{3} are in degrees 0, −1 respectively, we have

X_{−j}^{m} ⊗_{k𝔖_{m}} k = S^{m−j}M_{2} ⊗_{k} Λ^{j}M_{3} (j ≠ −1),
X_{−j}^{m} ⊗_{k𝔖_{m}} sgn = Λ^{m−j}M_{2} ⊗_{k} S^{j}M_{3} (j ≠ −1),
X_{1}^{m} ⊗_{k𝔖_{m}} k = S^{m}M_{1},
X_{1}^{m} ⊗_{k𝔖_{m}} sgn = Λ^{m}M_{1}.

Also we obtain the maps \overline{∂}_{−j} = ∂_{−j}⊗1 and \overline{∂′}_{−j} = ∂′_{−j}⊗1 upon taking tensor products. Effectively, under the identification, for any x_{1}, …, x_{m−j} ∈ M_{2} and y_{m−j+1}, …, y_{m} ∈ M_{3}, we have

\overline{∂}_{−j}(x_{1} ⋯ x_{m−j} ⊗ y_{m−j+1} ∧ ⋯ ∧ y_{m})
= ∑_{1≤i≤m−j} x_{1} ⋯ x̂_{i} ⋯ x_{m−j} ⊗ π(x_{i}) ∧ y_{m−j+1} ∧ ⋯ ∧ y_{m},

\overline{∂′}_{−j}(x_{1} ∧ ⋯ ∧ x_{m−j} ⊗ y_{m−j+1} ⋯ y_{m})
= ∑_{1≤i≤m−j}(−1)^{m−j−1}(x_{1} ∧ ⋯ ∧ x̂_{i} ∧ ⋯ ∧ x_{m−j} ⊗ π(x_{i})y_{m−j+1} ⋯ y_{m}),

here x̂_{i} means that x_{i} is missing from the terms.

Lemma 3. Let 0 → M_{1} → M_{2} → M_{3} → 0 be a short exact sequence of kG-modules. For each positive integer m, we have long exact sequences of kG-modules

0 → S^{m}M_{1} ⟶^{\overline{∂}_{1}} S^{m}M_{2} ⟶^{\overline{∂}_{0}} S^{m−1}M_{2} ⊗_{k} Λ^{1}(M_{3}) ⟶^{\overline{∂}_{−1}} ⋯ ⟶^{\overline{∂}_{−m}} Λ^{m}(M_{3}) → 0,

0 → Λ^{m}M_{1} ⟶^{\overline{∂′}_{1}} Λ^{m}M_{2} ⟶^{\overline{∂′}_{0}} Λ^{m−1}M_{2}⊗_{k}S^{1}(M_{3}) ⟶^{\overline{∂′}_{−1}} ⋯ ⟶^{\overline{∂′}_{−m}} S^{m}(M_{3}) → 0.

Proof. Let 0 ≤ j ≤ m. By standard algebra, we know that ker \overline{∂}_{−j} = (ker ∂_{−j}) ⊗_{k𝔖_{m}} k (see [2, §II.3.6, Proposition 6]). So that

ker \overline{∂_{−j}} = (im ∂_{−j+1}) ⊗_{k𝔖_{m}} k = im \overline{∂}_{−j+1}.

The map \overline{∂}_{1} is clearly injective. The exactness of the second sequence can be proved similarly. □

3. Main results

We are now ready to prove our main result. Before doing this, we deal with a simple case which basically states that the generic Jordan type of the exterior power (respectively, symmetric power) of a module depends solely on the generic Jordan types of the various exterior powers (respectively, symmetric powers) of its non-projective part.

In the following lemma, we use the convention that the value of \binom{r}{j} is 0 if j > r, and 1 if r = j = 0. Also, we use the convention that the value of 0^{j} is 0 if j > 0, and 1 if j = 0.

Lemma 4. Let M be a kE-module where E is an elementary abelian pgroup of order p^{n} and 1 ≤ m ≤ dim_{k} M be a positive integer. Suppose that, for each integer j such that 0 ≤ jp^{n} ≤ m, the stable generic Jordan type of Λ^{m−jp^{n}}Ω^{0}M (respectively, S^{m−jp^{n}}Ω^{0}M) is [1]^{a_{1,j}} ⋯ [p − 1]^{a_{p−1,j}}. Then the stable generic Jordan type of Λ^{m}M (respectively, S^{m}M) is [1]^{m_{1}} ⋯ [p − 1]^{m_{p−1}} where, for each 1 ≤ s ≤ p − 1, we have

m_{s} = ∑_{0≤jp^{n}≤m} a_{s,j}\binom{r}{j} (respectively, m_{s} = ∑_{0≤jp^{n}≤m} a_{s,j}r^{j})

and rp^{n} = dim_{k} M − dim_{k} Ω^{0}M. In particular, in the case of 1 ≤ m ≤ p^{n} − 1, the stable generic Jordan types of Λ^{m}M and Λ^{m}Ω^{0}M (respectively, S^{m}M and S^{m}Ω^{0}M) are identical.

Proof. Let M = Ω^{0}M ⊕ P where P is the largest free kE-submodule of M so that rp^{n} = dim_{k} P. It is well-known that

Λ^{m}M ≅ ⨁_{i=0}^{m} Λ^{m−i}Ω^{0}M ⊗_{k} Λ^{i}P.

By Lemma 1 (ii), Λ^{i}P is not generically free if and only if dim(V_{E}(Λ^{i}P)) = n if and only if i = jp^{n} for some non-negative integer j. In this case, by Lemma 1 (iii), the stable generic Jordan type of Λ^{i}P is [1]^{\binom{r}{j}}. So, upon taking tensor product, the stable generic Jordan type of Λ^{m−jp^{n}}Ω^{0}M ⊗_{k} Λ^{jp^{n}}P is

[1]^{a_{1,j} \binom{r}{j}} ⋯ [p − 1]^{a_{p−1,j} \binom{r}{j}}.

The case of symmetric power can be obtained similarly.

Suppose that 1 ≤ m ≤ p^{n} − 1. For the case of exterior power, we have m_{s} = a_{s,0}\binom{r}{0} = a_{s,0}. For the case of symmetric power, we have m_{s} = a_{s,0}r^{0} = a_{s,0}. □

Let M, N be kE-modules. We say that the stable generic Jordan types of M, N are complementary if the stable generic Jordan types of M, N are

[1]^{a_{1}}[2]^{a_{2}} ⋯ [p − 2]^{a_{p−2}}[p − 1]^{a_{p−1}},
[1]^{a_{p−1}}[2]^{a_{p−2}} ⋯ [p − 2]^{a_{2}}[p − 1]^{a_{1}}

respectively.

Theorem 5. Let E be an elementary abelian p-group of order p^{n} and M be a kE-module. Let m be a positive integer such that 1 ≤ m ≤ p^{n} − 1.

(i) Suppose that m is even. Let i, j be any integers. Then the stable generic Jordan types of Λ^{m}Ω^{2i}M and S^{m}Ω^{2j+1}M are identical. Similarly, the stable generic Jordan types of S^{m}Ω^{2i}M and Λ^{m}Ω^{2j+1}M are identical.

(ii) Suppose that m is odd. Then the stable generic Jordan types of Λ^{m}Ω^{2i}M (respectively, S^{m}Ω^{2i+1}M, Λ^{m}Ω^{2i+1}M and S^{m}Ω^{2i}M) are identical for all integers i. Furthermore, for any integers i, j, the stable generic Jordan types of Λ^{m}Ω^{2i}M, S^{m}Ω^{2j+1}M (respectively, Λ^{m}Ω^{2i+1}M, S^{m}Ω^{2j}M) are complementary.

In both cases, the stable generic Jordan types of Λ^{m}M and Λ^{m}Ω^{0}M (respectively, S^{m}M and S^{m}Ω^{0}M) are identical.

Proof. Let P_{M} be the projective cover of M so that we have a short exact sequence 0 → ΩM → P_{M} → M → 0. Consider the long exact sequence obtained in Lemma 3,

0 → S^{m}ΩM ⟶^{ϕ_{m+1}} S^{m}P_{M} ⟶^{ϕ_{m}} S^{m−1}P_{M} ⊗_{k} Λ^{1}M → ⋯ ⟶^{ϕ_{1}} Λ^{m}M → 0.

Since P_{M} is a free kE-module, we obtain that, by Lemma 1 (ii), S^{i}P_{M} are generically free and thus the kE-modules S^{i}P_{M} ⊗_{k} Λ^{m−i}M are generically free for all 1 ≤ i ≤ m. By using the short exact sequence, for 2 ≤ i ≤ m + 1,

0 → ker ϕ_{i} → S^{i}P_{M} ⊗_{k} Λ^{m−i}M → ker ϕ_{i−1} → 0

and arguing inductively, we obtain that the stable generic Jordan types of Λ^{m}M, S^{m}ΩM are identical if m is even, and complementary if m is odd. Similarly, by applying another long exact sequence of Lemma 3, we obtain that the stable generic Jordan types of S^{m}M, Λ^{m}ΩM are identical if m is even, and complementary if m is odd. Repeat the argument with the short exact sequence 0 → M → I_{M} → Ω^{−1}M → 0 and using the fact that ΩΩ^{−1} ≃ Ω^{0} ≃ Ω^{−1}Ω, we have all the desired results. The final statement is precisely given by Lemma 4. □

Specializing to the case of the trivial module k, we can describe the stable generic Jordan types of the exterior and symmetric powers of the various Heller translations of k.

Corollary 6. Let k be the trivial kE-module where E is an elementary abelian p-group of order p^{n}. Let 1 ≤ m ≤ p^{n} − 1 and i be an integer. Then the kE-module Λ^{m}Ω^{i}k has stable generic Jordan type

∅  if i is even and m ≥ 2,
[1]  if i is odd and m is even, or i is even and m = 1,
[p − 1] if both m, i are odd;

meanwhile, the kE-module S^{m}Ω^{i}k has stable generic Jordan type

∅  if i is odd and m ≥ 2,
[1]  if i is even,
[p − 1] if i is odd and m = 1.

References

[1] D. J. Benson, Representations and Cohomology I, II, 2nd edn. Cambridge Studies in Advanced Mathematics 30, 31, Cambridge University Press, Cambridge, 1990.

[2] N. Bourbaki, Algebra (Chapters 13), Springer, Berlin, 1989.

[3] J. F. Carlson, E. M. Friedlander and J. Pevtsova, ‘Modules of constant Jordan type’, J. Reine Angew. Math. 614 (191-234), 2008.

[4] L. Evens, The Cohomology of Groups, Oxford University Press, 1991.

[5] E. M. Friedlander, J. Pevtsova and A. Suslin, ‘Generic and maximal Jordan types’, Invent. Math. 168 (485–522), 2007.

[6] W. W. Wheeler, ‘Generic Module Theory’, J. Algebra, 185 (205–228), 1996.

(D. J. Benson) Institute of Mathematics, University of Aberdeen, King’s College, Fraser Noble Building, Aberdeen, AB24 3UE, United Kingdom.

E-mail address: d.j.benson@abdn.ac.uk

(K. J. Lim) Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076.

Current address: Division of Mathematical Sciences, Nanyang Technological University, SPMS-MAS-03-01, 21 Nanyang Link, Singapore 637371.

E-mail address: limkj@ntu.edu.sg