THE COMPLEXITY OF THE SPECHT MODULES CORRESPONDING TO HOOK PARTITIONS

KAY JIN LIM

Abstract. We show that the complexity of the Specht module corresponding to any hook partition is the p-weight of the partition. We calculate the variety and the complexity of the signed permutation modules. Let E_{s} be a representative of the conjugacy class containing an elementary abelian p-subgroup of a symmetric group generated by s disjoint p-cycles. We give formulae for the generic Jordan types of signed permutation modules restricted to E_{s} and of Specht modules corresponding to hook partitions μ restricted to E_{s} where s is the p-weight of μ.

1. Introduction

Alperin and Evens [1] introduced the complexity of finitely generated modules over finite group algebras. Meanwhile, Carlson [4] had been studying some varieties for finitely generated modules over finite group algebras. Over elementary abelian p-groups, he showed that the complexity of a module is precisely the dimension of the cohomological variety and the dimension of the rank variety corresponding to the module. Avrunin and Scott [2] proved an analogue of Quillen’s Stratification. The complexity of a kG-module can be determined by looking at the restriction of the module to the elementary abelian p-subgroups of the group G. The study of the varieties for modules over elementary abelian p-groups is closely related to the study of the notion of generic Jordan types of the modules introduced by Wheeler [14].

A partition μ = (μ_{1}^{a_{1}}, …, μ_{s}^{a_{s}}) is p × p if for each 1 ≤ i ≤ s, both μ_{i} and a_{i} are multiples of p. The VIGRE group in Georgia made the following conjecture.

Conjecture 1.1 (VIGRE 2004). The complexity of the Specht module S^{μ} is the p-weight of the partition μ if and only if μ is not p × p.

In [13], we studied the support varieties and the complexities for Specht modules corresponding to some p-regular partitions and the partition (p^{p}). We showed that, for the case of abelian defects, Conjecture 1.1 is true. We also showed that a large class of Specht modules satisfy Conjecture 1.1. Recently [11], Hemmer shows one direction of Conjecture 1.1, i.e., if a partition μ is p × p, then the complexity of the Specht module S^{μ} is strictly less than the p-weight of μ.

Let E be an elementary abelian p-group of rank m. To show that a kE-module M has complexity m, it suffices to show that the module M is not generically free (see Proposition 2.2). Let μ be a hook partition of n, s be the p-weight of μ and E_{s} be the elementary abelian p-subgroup of 𝔖_{n} generated by the p-cycles ((i − 1)p + 1, (i − 1)p + 2, …, ip) with 1 ≤ i ≤ s. We consider the restricted module S^{μ}↓_{E_{s}} and show that the module is not generically free. This implies that the complexity of the Specht module S^{μ} is bounded below by s. Since the value s is also the p-rank of a defect group of the block containing S^{μ}, we get the other half of Conjecture 1.1 for hook partitions.

Remark 1.2. Following 4.1 of [13], one may wonder if V_{𝔖_{n}}(S^{μ}) = res_{𝔖_{,}D_{μ}}^{∗} V_{D_{μ}}(k) whenever μ is not p × p. A counterexample to the above statement is the partition μ = (7, 1^{3}) where the vertex of S^{μ} is C_{3} × C_{3} × C_{3} [15] and Dμ̃ ≅ C_{3} ≀ C_{3}.

Theorem 1.3. For any hook partition μ, the complexity of the Specht module S^{μ} is exactly the p-weight of μ.

2. Background materials and notations

Most of the basic materials about group cohomology and the representation theory of symmetric groups can be found in [3] and [10] respectively.

Let G be a finite group, k be an algebraically closed field of characteristic prime p and Ext_{kG}^{∗}(M, M) be the cohomology ring of a finitely generated kG-module M. The cohomological variety V_{G}(M) for the module M is the set of maximal ideals spectrum of Ext_{kG}^{•}(k, k) containing the kernel of the map Φ_{M} : Ext_{kG}^{•}(k, k) ⟶^{⊗_{k}M} Ext_{kG}^{∗}(M, M) where

Ext_{kG}^{•}(k, k) = \{ \begin{matrix}Ext_{kG}^{ev}(k, k) & p is odd \\ Ext_{kG}^{∗}(k, k) & p = 2. \end{matrix}

We have Theorem [2]

V_{G}(M) = ⋃_{E∈E(G)} res_{G,E}^{∗} V_{E}(M)

where E(G) is a set of representatives for the conjugacy classes of elementary abelian p-subgroups of G and res_{G,E}^{∗} : V_{E}(k) → V_{G}(k) is the map induced by the restriction res_{G,E} : Ext_{kG}^{•}(k, k) → Ext_{kE}^{•}(k, k). So dim V_{G}(M) = max_{E∈E(G)}{dim V_{E}(M)}. Let E be an elementary abelian p-group of rank n generated by the elements g_{1}, g_{2}, …, g_{n}. The rank variety V_{E}^{♯}(M) of a finitely generated kE-module M is the set

{0 ≠ α ∈ k^{n} | M↓_{k⟨u_{α}⟩} is not free} ∪ {0}

where u_{α} = 1 + ∑_{i=1}^{n} α_{i}(g_{i} − 1) and α = (α_{1}, …, α_{n}) ∈ k^{n}. Avrunin and Scott [2] showed that V_{E}^{♯}(M) ≅ V_{E}(M).

Let α be a generic point in k^{n}. The generic Jordan type of a finitely generated kE-module M is the Jordan type of the restricted module M↓_{⟨u_{α}⟩} where ⟨u_{α}⟩ is the cyclic group C_{p} of order p described above [14]. In this case, we write (1^{n_{1}}, …, p^{n_{p}}) for the generic Jordan type of M if the number of Jordan type of size 1 ≤ i ≤ p is n_{i}. Suppose that β is a multiple of α. We have M↓_{⟨u_{β}⟩} ≅ M↓_{⟨u_{α}⟩}. We write [α]^{∗}(M) for the isomorphism class of kC_{p}-modules containing M↓_{⟨u_{α}⟩}. We now state a partial result of 4.7 [8] for our case.

Proposition 2.1 (Part of 4.7 [8]). Let E be an elementary abelian p-group, M and N be finitely generated kE-modules and α ∈ k^{n} be a generic point. We have [α]^{∗}(M ⊕ N) ≅ [α]^{∗}(M) ⊕ [α]^{∗}(N).

The stable generic Jordan type of M is the generic Jordan type of M modulo its projective summands. With the notation introduced earlier, the stable generic Jordan type of M is (1^{n_{1}}, …, (p − 1)^{n_{p−1}}).

Proposition 2.2. Let E be an elementary abelian p-group of rank n. A kE-module M is not generically free if and only if V_{E}^{♯}(M) = V_{E}^{♯}(k).

Proof. Suppose that V_{E}^{♯}(M) has dimension m < n. Let α ∈ k^{n} be a generic point and l_{α} ⊆ k^{n} be the line containing the point α. Note that V_{E}^{♯}(M) is a closed homogeneous affine variety (see Theorem 4.3 of [5]). Since m + 1 ≤ n, we have V_{E}^{♯}(M) ∩ l_{α} = {0}, i.e., M is generically free. □

Let n be a natural number. A partition μ of n is a non-increasing sequence of positive integers (μ_{1}, …, μ_{s}) such that ∑_{i=1}^{s} μ_{i} = n. In this case, we write n = |μ|. The Young diagram [μ] is the subset of ℤ^{2} consisting of all nodes (i, j) satisfying 1 ≤ i ≤ s and 1 ≤ j ≤ μ_{i}. The p-core μ̃ of μ is the partition corresponding to the Young diagram obtained from [μ] by removing as many skew p-hooks as possible. The number of skew p-hooks removed from [μ] to get [μ̃] is called the p-weight of μ. A μ-tableau is an assignment of the numbers 1, 2, …, n to the nodes of the Young diagram [μ]. For each (i, j) ∈ [μ], we write t_{ij} for the number assigned to (i, j). For each number 1 ≤ m ≤ n, we write t_{m} for the node (i, j) such that t_{ij} = m. We write R_{i}(t) for the set consisting of the numbers in the ith row of t, i.e., R_{i}(t) = {t_{ij} | j ≥ 1}. Similarly, we write C_{j}(t) for the set consisting of the numbers in the jth column of t. If numbers are increasing down each column and along each row of t, we say that t is a standard μ-tableau.

Let 𝔖_{n} denote the symmetric group on n letters. The group acts on the set of all μ-tableaux t by permuting the numbers assigned to t. Let R_{t} and C_{t} be the row stabilizer and column stabilizer of t respectively, i.e., R_{t} = 𝔖_{R_{1}(t)} × … × 𝔖_{R_{s}(t)} and C_{t} = 𝔖_{C_{1}(t)} × … × 𝔖_{C_{μ_{1}}(t)} where 𝔖_{Ω} is the symmetric group corresponding to a given set Ω. We define an equivalence relation on the set of all μ-tableaux, s ∼ t if and only if s = σt for some σ ∈ R_{t}. A μ-tabloid {t} is the equivalence class containing the μ-tableau t. The μ-polytabloid corresponding to the μ-tableau t is

e_{t} = ∑_{σ∈C_{t}} sgn(σ){σt}.

The k-vector space spanned by all μ-polytabloids forms a k𝔖_{n}-module. It is called the Specht module S^{μ}. If t is a standard μ-tableau, the μ-polytabloid e_{t} is called a standard μ-polytabloid. The set of all standard μ-polytabloids forms a basis of S^{μ}, the standard basis of S^{μ}. The Young subgroup 𝔖_{μ} of 𝔖_{n} corresponding to μ is

𝔖_{{1,...,μ_{1}}} × 𝔖_{{μ_{1}+1,...,μ_{1}+μ_{2}}} × … × 𝔖_{{μ_{1}+...+μ_{s−1}+1,...,μ_{1}+...+μ_{s}}}.

The signed permutation module corresponding to a pair of partitions (α, β) with |α| = a and |β| = b is the induced module

M(α|β) = (k ⊠ sgn)↑_{𝔖_{α}×𝔖_{β}}^{𝔖_{a+b}}

where ⊠ denotes the exterior tensor product of two modules [6]. It generalizes the notion of permutation modules, by taking b = 0.

For each 1 ≤ s ≤ ⌊n/p⌋, we write E_{s} for the elementary abelian p-subgroup of 𝔖_{n} generated by the p-cycles g_{i} = ((i − 1)p + 1, (i − 1)p + 2, …, ip) with 1 ≤ i ≤ s. For each positive integer i, we write I_{i} for the set {(i − 1)p + 1, (i − 1)p + 2, …, ip}.

Proposition 2.3. Let μ be a partition of n = dp + r with 0 ≤ r ≤ p − 1.

(i) [The Branching Theorem §9 of [9]] Let Ω(μ) be the set of partitions of n − 1 obtained from μ by removing a node. The module S^{μ}↓_{𝔖_{n−1}} has a Specht filtration with Specht factors S^{λ} one for each λ ∈ Ω(μ).

(ii) [Nakayama’s Conjecture] Let λ be another partition of n. The Specht modules S^{μ}, S^{λ} lie in the same block if and only if the corresponding partitions μ, λ have the same p-cores.

(iii) If |μ̃| > r, then S^{μ}↓_{E_{d}} is generically free.

Proof. For a proof of Nakayama’s Conjecture, see §6 of [10]. The proof of (iii) is similar to the proof of Proposition 2.2 (iv) [13]. Since the p-weight m_{μ} of μ is strictly less than d, a defect group D_{μ} of the block containing S^{μ} has p-rank m_{μ} and V_{𝔖_{n}}(S^{μ}) ⊆ res_{𝔖_{n},D_{μ}}^{∗} V_{D_{μ}}(S^{μ}) (Proposition 2.1 (iv) of [13]), we have dim V_{E_{d}}(S^{μ}) ≤ m_{μ} < d. So S^{μ}↓_{E_{d}} is generically free by Proposition 2.2. □

3. Signed permutation modules

Let n_{1}, n_{2}, …, n_{u} and s be non-negative integers. We define the set Λ(n_{1}, …, n_{u}; s) to consist of all u-tuples (c_{1}, …, c_{u}) ∈ (ℕ_{≥0})^{u} such that 0 ≤ c_{i} ≤ n_{i} for each 1 ≤ i ≤ u and c_{1}+c_{2}+…+c_{u} = s. If s > ∑_{i=1}^{u} n_{i} or s < 0, then Λ(n_{1}, …, n_{u}; s) = ∅.

Theorem 3.1. Let α = (α_{1}, …, α_{m}), β = (β_{1}, …, β_{n}) and |α| + |β| = dp + r with 0 ≤ r ≤ p − 1. Suppose that the p-residue of α_{i} is s_{i} for each 1 ≤ i ≤ m, the p-residue of β_{j} is s_{m+j} for each 1 ≤ j ≤ n and ∑_{i=1}^{m+n} s_{i} = cp + r. Let 1 ≤ s ≤ d.

(i) The stable generic Jordan type of M(α|β)↓_{E_{s}} is (1^{N(α,β,s)}) where

N(α, β, s) = ∑_{(c_{1},...,c_{m+n})∈Λ} (\frac{s!}{∏_{i=1}^{m+n} c_{i}!} · \frac{((d − s)p + r)!}{∏_{i=1}^{m}(α_{i} − c_{i}p)! ∏_{j=1}^{n}(β_{j} − c_{j+m}p)!})

where Λ = Λ (\frac{α_{1}−s_{1}}{p} , …, \frac{α_{m}−s_{m}}{p} , \frac{β_{1}−s_{m+1}}{p} , …, \frac{β_{n}−s_{m+n}}{p}; s). The empty sum is defined to be zero. In this case, the module is generically free.

(ii) We have V_{𝔖_{|α|+|β|}}(M(α|β)) = res_{𝔖_{|α|+|β|},𝔖_{α}×𝔖_{β}}^{∗} V_{𝔖_{α}×𝔖_{β}}(k). In particular, the complexity of the signed permutation module M(α|β) is d − c.

Proof. We use the Mackey decomposition formula (see 3.3.4 [3]),

(k ⊠ sgn)↑_{𝔖_{α}×𝔖_{β}}^{𝔖_{|α|+|β|}}↓_{E_{s}} ≅ ⨁_{E_{s}g(𝔖_{α}×𝔖_{β})} ^{g}(k ⊠ sgn)↓_{E_{s}∩^{g}(𝔖_{α}×𝔖_{β})}↑^{E_{s}}.

If E_{s} ∩^{g}(𝔖_{α} ×𝔖_{β}) ⪇ E_{s}, then ^{g}(k ⊠sgn)↓_{E_{s}∩^{g}(𝔖_{α}×𝔖_{β})}↑^{E_{s}} is generically free. Suppose that E_{s}∩^{g}(𝔖_{α}×𝔖_{β}) = E_{s}, i.e., E_{s}g ⊆ g(𝔖_{α}×𝔖_{β}). The double coset representatives of the subgroups E_{s}, 𝔖_{α} × 𝔖_{β} in 𝔖_{|α|+|β|} correspond to the orbits of the μ-tabloids under the action of E_{s} where μ = (α_{1}, …, α_{m}, β_{1}, …, β_{n}). So the number of double coset representatives fixed by E_{s} is precisely the number of μ-tabloids fixed by E_{s}.

A μ-tabloid {t} is fixed by E_{s} if and only if for each 1 ≤ i ≤ s, we have I_{i} ⊆ R_{j(i)}(t) for some 1 ≤ j(i) ≤ m + n. In this case, it is necessary that

∑_{i=1}^{m}(α_{i} − s_{i}) + ∑_{j=1}^{n}(β_{j} − s_{m+j}) ≥ sp

i.e., s ≤ d − c. So Λ = ∅ if and only if s > d − c. Suppose that s ≤ d − c. We fix an element (c_{1}, c_{2}, …, c_{m+n}) in the set Λ. The number of choices assigning each I_{i} with 1 ≤ i ≤ s into a row of the partition (c_{1}p, c_{2}p, …, c_{m+n}p) is

\frac{s!}{∏_{i=1}^{m+n} c_{i}!}.

Independently, the number of choices assigning the remaining (d − s)p + r numbers sp + 1, sp + 2, …, dp + r into the remaining (d − s)p + r nodes of [μ] with μ_{i} − c_{i}p nodes in each ith row is

\frac{((d − s)p + r)!}{∏_{i=1}^{m}(α_{i} − c_{i}p)! ∏_{j=1}^{n}(β_{j} − c_{j+m}p)!}.

If we sum up over all elements of Λ, we get N(α, β, s). In these cases, the generic Jordan type of ^{g}(k ⊠ sgn)_{E_{s}} is (1). This completes the proof for (i).

Let G = 𝔖_{|α|+|β|} and H = 𝔖_{α} × 𝔖_{β}. By Proposition 8.2.4 of [7] and Proposition 2.1 (iii) [13], we have V_{G}(M(α|β)) = res_{G,H}^{∗} V_{H}(k ⊠ sgn) = res_{G,H}^{∗} V_{H}(k). Since the map res^{∗} is a finite map (4.2.5 of [3] II), we have

dim V_{G}(M(α|β)) = dim V_{H}(k)

= p-rank of H

= ∑_{i=1}^{m}(α_{i} − s_{i})/p + ∑_{j=1}^{n}(β_{j} − s_{j+m})/p

= d − c  □

Remark 3.2. Theorem 3.1 (ii) is an obvious generalization of 3.2.2 [12].

4. Proof of Theorem 1.3

The proof of Theorem 1.3 is a consequence of a more general statement, which we explicitly compute the stable generic Jordan type of S^{μ}↓_{E_{s}} where s is the p-weight of a hook partition μ (Corollary 4.2 and Theorem 4.5). Let μ = (a, 1^{b}). Our aim is to show that S^{μ}↓_{E_{s}} is not generically free. We consider two cases, p ∤ a + b and p | a + b. We shall briefly describe the proofs of Theorem 4.1 and Theorem 4.5.

By the Littlewood-Richardson Rule (see 2.8.13 of [10]), the signed permutation module M((a)|(b)) has a Specht filtration with Specht factors S^{(a,1^{b})} and S^{(a+1,1^{b−1})}. In the case where a + b ≢ 0(mod p), the sizes of p-cores are nonzero and b ≢ b + 1(mod p), so p-cores of (a, 1^{b}), (a + 1, 1^{b−1}) are distinct. Using Nakayama’s Conjecture, we have a direct sum decomposition M((a)|(b)) ≅ S^{(a,1^{b})} ⊕ S^{(a+1,1^{b−1})}. We prove Theorem 4.1 by using induction on b.

The case for a + b ≡ 0(mod p) is slightly more complicated. We show that the short exact sequence

0 → S^{(a−1,1^{b+1})} → S^{(a,1^{b+1})} → S^{(a,1^{b})} → 0

generically splits, i.e., S^{(a,1^{b+1})}↓_{⟨u_{α}⟩} ≅ S^{μ}↓_{⟨u_{α}⟩} ⊕ S^{λ}↓_{⟨u_{α}⟩} for a generic point α ∈ k^{d} where a + b = dp. With Corollary 4.2, we prove Theorem 4.5 by using induction on b.

4.1. Hook of size not a multiple of p.

Theorem 4.1. Let μ = (a, 1^{b}), a + b = dp + r, a = up + a_{0} and b = vp + b_{0} with 0 ≤ r, a_{0}, b_{0} ≤ p − 1 and r ≠ 0. For any 1 ≤ s ≤ d, the stable generic Jordan type of S^{μ}↓_{E_{s}} is (1^{N(μ;s)}) where

N(μ; s) = ∑_{(c_{1},c_{2})∈Λ(u,v;s)} \binom{s}{c_{2}}\binom{(d − s)p + r − 1}{b − c_{2}p}.

Proof. If d = 0, there is nothing to prove. For any hook (a, 1^{b}), we write Λ((a, 1^{b}); s) for the set Λ(u, v; s). We now fix the numbers a, b. Let μ = (a, 1^{b}) and λ = (a − 1, 1^{b+1}). We prove the formula by using induction on the number b. If b = 0, then S^{μ} is the trivial module and it has Jordan type (1). On the other hand, the set Λ(d, 0; s) contains precisely one element (s, 0). So N(μ; s) = \binom{s}{0}\binom{(d−s)p+r−1}{0} = 1 given that r ≥ 1. Suppose that for some 0 ≤ b, the module S^{μ}↓_{E_{s}} has stable generic Jordan type as given by the formula. Since r ≠ 0, we have a direct sum decomposition M((a−1)|(b+1)) ≅ S^{μ}⊕S^{λ}. Let a_{1}, b_{1} be the p-residues of a−1, b+1 respectively. It is clear that b_{1} ≡ b_{0}+1(mod p) and a_{0} ≡ a_{1}+1(mod p). By Theorem 3.1, M((a − 1)|(b + 1))↓_{E_{s}} has stable generic Jordan type (1^{N((a−1),(b+1),s)}) where

N((a − 1), (b + 1), s) = ∑_{(c_{1},c_{2})∈Λ(λ;s)} \binom{s}{c_{2}}\binom{(d − s)p + r}{b + 1 − c_{2}p}.

Our aim is to show that N(λ; s) = N((a − 1), (b + 1), s) − N(μ; s). We consider 4 cases. Case (i): Suppose that r ≤ b_{0} ≤ b_{1}. If b_{0} > r, then Λ(λ; s) = Λ(μ; s) and c = 1. The stable generic Jordan type of S^{λ}↓_{E_{s}} is (1^{w}) where

w = ∑_{(c_{1},c_{2})∈Λ(λ;s)} \binom{s}{c_{2}}\binom{(d − s)p + r}{b + 1 − c_{2}p} − ∑_{(c_{1},c_{2})∈Λ(μ;s)} \binom{s}{c_{2}}\binom{(d − s)p + r − 1}{b − c_{2}p}

= ∑_{(c_{1},c_{2})∈Λ(λ;s)} \binom{s}{c_{2}} (\binom{(d − s)p + r}{b + 1 − c_{2}p} − \binom{(d − s)p + r − 1}{b − c_{2}p})

= ∑_{(c_{1},c_{2})∈Λ(λ;s)} \binom{s}{c_{2}}\binom{(d − s)p + r − 1}{b + 1 − c_{2}p} = N(λ; s)

Suppose that b_{0} = r, we have Λ(λ; s) ∪ {(u, s − u)} = Λ(μ; s) if s < d; otherwise, Λ(λ; s) = ∅. In the first case, \binom{(d−s)p+r−1}{b−(s−u)p} = 0 given that b − (s − u)p = vp + r − (s − u)p = (d − s)p + r > (d − s)p + r − 1. So the stable generic Jordan type of S^{λ}↓_{E_{s}} is (1^{w}) = (1^{N(λ;s)}). In the second case, the module M((a − 1)|(b + 1))↓_{E_{s}} is generically free. As a direct summand of M((a − 1)|(b + 1))↓_{E_{s}}, the module S^{λ}↓_{E_{s}} is generically free. This fits into the formula. Case (ii): Suppose that r ≤ b_{0} and b_{1} < r, i.e., b_{1} = 0 and b_{0} = p − 1. Let a_{0} > 0. We have Λ(λ; s) = Λ(μ; s) ∪ {(s − v − 1, v + 1)} if s ≥ v + 1; otherwise, Λ(λ; s) = Λ(μ; s). The second case is easy. For the first case, we have an extra term in N((a − 1), (b + 1), s) which is \binom{s}{v+1}\binom{(d−s)p+r}{b+1−(v+1)p} = \binom{s}{v+1}; on the other hand, we also have an extra term in N(λ; s) which is \binom{s}{v+1}\binom{(d−s)p+r−1}{b+1−(v+1)p} = \binom{s}{v+1}. This shows the desired formula for the stable generic Jordan type of S^{λ}↓_{E_{s}}. Let a_{0} = 0. In this case r = p − 1. So we have Λ(λ; s) = (Λ(μ; s) − {(u, s − u)}) ∪ {(s − v − 1, v + 1)} if s ≥ v+1; otherwise, Λ(λ; s) = Λ(μ; s)−{(u, s−u)}. For the extra element (u, s−u), we have b − (s − u)p = vp + (p − 1) − sp + up = (d − s)p + (p − 1) > (d − s)p + r − 1. So the extra term corresponding to (u, s − u) is superfluous in N(μ; s). Now the inductive argument follows similarly as the case where a_{0} > 0. Case (iii): Suppose that b_{0} ≤ b_{1} < r. In this case, Λ(λ; s) = Λ(μ; s). So the inductive step is easy, the stable generic Jordan type of S^{λ}↓_{E_{s}} is (1^{N(λ;s)}). Case (iv): Suppose that b_{0} < r and r ≤ b_{1}, i.e., b_{0} = r − 1 and b_{1} = r. In this case, Λ(λ; s) = Λ(μ; s). So the stable generic Jordan type of S^{λ}↓_{E_{s}} is (1^{N(λ;s)}). □

Corollary 4.2. Let μ = (a, 1^{b}), a + b = dp + r with d ≠ 0, 1 ≤ r ≤ p − 1 and b = vp + b_{0} with 0 ≤ b_{0} ≤ p − 1.

(i) If r ≤ b_{0}, then V_{E_{d−1}}^{♯}(S^{μ}) = V_{E_{d−1}}^{♯}(k) and the complexity of S^{μ} is d − 1. In

this case, the stable generic Jordan type of S^{μ}↓_{E_{d−1}} is (1^{N(μ;d−1)}) with

N(μ; d − 1) = \binom{d − 1}{v}\binom{p + r − 1}{b_{0}} ≠ 0.

(ii) If b_{0} < r, then V_{E_{d}}^{♯}(S^{μ}) = V_{E_{d}}^{♯}(k) and the complexity of S^{μ} is d. In this case, the stable generic Jordan type of S^{μ}↓_{E_{d}} is (1^{N(μ;d)}) with

N(μ; d) = \binom{d}{v}\binom{r − 1}{b_{0}} ≠ 0.

Proof. Suppose that d ≥ 1; otherwise, the result is trivial. In general, the complexity of an indecomposable kG-module is bounded above by the p-rank of a defect group of the block containing the module (see 2.1 (iv) of [13]). For b_{0} ≥ r, a defect group of the block containing S^{μ} is the Sylow p-subgroup of the symmetric group 𝔖_{(d−1)p}, it has p-rank d − 1. Consider S^{μ}↓_{E_{d−1}} and apply the formula in Theorem 4.1, we have

N(μ; d − 1) = ∑_{(c_{1},c_{2})∈Λ(u,v;d−1)} \binom{d − 1}{c_{2}}\binom{p + r − 1}{b − c_{2}p}

= \binom{d − 1}{v}\binom{p + r − 1}{b_{0}}.

Note that N(μ; d − 1) ≠ 0 unless v = d and b_{0} = r, i.e., a = 0. For b_{0} < r, a defect group of the block containing S^{μ} is the Sylow p-subgroup of the symmetric group 𝔖_{dp}. Apply Theorem 4.1 with s = d, we have

N(μ; d) = ∑_{(c_{1},c_{2})∈Λ(u,v;d)} \binom{d}{c_{2}}\binom{r − 1}{b − c_{2}p} = \binom{d}{v}\binom{r − 1}{b_{0}} ≠ 0. □

4.2. Hook of size a multiple of p. Let μ, λ be partitions of n. We write λ ⊴ μ if μ dominates λ by the total ordering. In the case where λ ⊴ μ and λ ≠ μ, we write λ ◁ μ. Let μ be a hook partition with |μ| ≥ p and t be a μ-tableau. We associate t to a partition λ(t) = (u, 1^{p−u}) of p where u = |R_{1}(t) ∩ I_{1}|.

Lemma 4.3. Let μ be a hook partition with |μ| ≥ p and t be a standard μ-tableau.

(i) The standard μ-polytabloids e_{s} involved in g_{1}e_{t} satisfy the ordering λ(s) ⊴ λ(t). In the case λ(s) = λ(t), we have R_{1}(s) − I_{1} = R_{1}(t) − I_{1}. In the case λ(s) ◁ λ(t), we have (R_{1}(t) − I_{1}) ∪ {m} = R_{1}(s) − I_{1} for some number m ∈ C_{1}(t).

(ii) For 2 ≤ i ≤ ⌊|μ|/p⌋, g_{i} permutes the standard μ-polytabloids up to a sign, R_{1}(g_{i}t) − I_{i} = R_{1}(t) − I_{i} and λ(g_{i}t) = λ(t). Furthermore, g_{i}e_{t} = ±e_{t} if and only if I_{i} ⊆ R_{1}(t) or I_{i} ⊆ C_{1}(t). In this case, g_{i}e_{t} = e_{t}.

Proof. All we need are the Garnir relations. Consider g_{1}e_{t} = e_{g_{1}t}. Note that e_{g_{1}t} = εe_{w} where ε = ±1 and w is the μ-tableau obtained from g_{1}t by first rearranging numbers in the first row of g_{1}t except 2 and then numbers in the first column of g_{1}t such that numbers are increasing along the first row ignoring the first node and numbers are increasing down the first column. Note that R_{1}(w)−I_{1} = R_{1}(t)−I_{1}. If 1 ∈ C_{1}(w), then w is standard. In this case, λ(w) = λ(t) and R_{1}(w)−I_{1} = R_{1}(t)−I_{1}. Suppose that 1 ∉ C_{1}(w), i.e., w_{12} = 1. We use the Garnir relation for the first two columns of w. Consider the left coset representatives (1, u) with u ∈ C_{1}(w) ∪ {1} of 𝔖_{C_{1}(w)} × 𝔖_{{1}} in 𝔖_{C_{1}(w)∪{1}}. So

g_{1}e_{t} = ε ∑_{m∈C_{1}(w)} e_{(1m)w}.

Note that for each m ∈ C_{1}(w), e_{(1m)w} is standard up to a sign. If m ∈ C_{1}(w) ∩ I_{1}, then λ((1m)w) = λ(t) and R_{1}((1m)w) − I_{1} = R_{1}(w) − I_{1} = R_{1}(t) − I_{1}. If m ∈ C_{1}(w) − I_{1} ⊆ C_{1}(t), then λ((1m)w) ◁ λ(t) and R_{1}((1m)w) − I_{1} = (R_{1}(w) − I_{1}) ∪ {m} = (R_{1}(t) − I_{1}) ∪ {m}. Case (i) is established.

Suppose that 2 ≤ i ≤ ⌊|μ|/p⌋. Note that (g_{i}t)_{11} = 1. Let σ ∈ C_{1}(g_{i}t) such that σ(g_{i}t) has numbers increasing down the first column. So sgn(σ)e_{g_{i}t} is a standard polytabloid. Since g_{i}(R_{1}(t) ∩ I_{i}) = R_{1}(g_{i}t) ∩ I_{i}, this gives the equivalent statement. If I_{i} ⊆ R_{1}(t), clearly g_{i}e_{t} = e_{t}. In the case I_{i} ⊆ C_{1}(t), we have g_{i}e_{t} = sgn(g_{i})e_{t} = e_{t}. □

Recall that for any point α = (α_{1}, …, α_{s}) ∈ k^{s} we define the element u_{α} = 1 + ∑_{i=1}^{s} α_{i}(g_{i} − 1) in the group algebra kE_{s}.

Lemma 4.4. Let μ = (a, 1^{b}), a+b = dp+r with 0 ≤ r ≤ p−1. Let 1 ≤ s ≤ d, α ∈ k^{s} be a generic point and t be a standard μ-tableau. Fix an integer 1 ≤ m ≤ p − 1. If (u_{α} − 1)^{m}e_{t} = 0, then for any 2 ≤ j ≤ s the set I_{j} lies entirely inside either the first column of μ or the first row of μ.

Proof. Suppose that there is some 2 ≤ j ≤ s such that I_{j} ⊈ R_{1}(t) or I_{j} ⊈ C_{1}(t). By Lemma 4.3 (ii), g_{j} permutes the set of μ-standard polytabloids up to a sign. The size of the orbit O(e_{t}) under the action of g_{j} is p, up to a sign. Note that (u_{α} − 1)^{m} is a linear combination of some products of not more than m copies of g_{i}’s with 1 ≤ i ≤ s (may be repeated). Fix an m-string β = (β_{1}, …, β_{m}) with each β_{i} ∈ {0, 1, …, s}, we write g_{β} = g_{β_{1}}g_{β_{2}} … g_{β_{m}} assuming that g_{β_{0}} = 1. Note that λ(g_{j}^{m}t) = λ(t). We claim that g_{β}e_{t} does not involve g_{j}^{m}e_{t} up to a sign unless and only unless β = j = (j, …, j). Once we have proved this claim, since g_{j} occurs precisely once with coefficient α_{j}^{m} in the expansion of (u_{α} − 1)^{m} and the point α is generic, we conclude that (u_{α} − 1)^{m}e_{t} ≠ 0.

For each 1 ≤ i ≤ d, let R_{1,i}(s) = R_{1}(s) ∩ I_{i} for a μ-tableau s. Let h, l be the multiplicities of g_{1}, g_{j} appearing in g_{β} respectively. Let e_{s} be a standard polytabloid involved in g_{β}e_{t} such that λ(s) = λ(t). By Lemma 4.3 (i) and (ii), we have R_{1}(s) − I_{1} = (g_{β}g_{1}^{−h})(R_{1}(t) − I_{1}) and R_{1,j}(s) = g_{j}^{l}R_{1,j}(t). So R_{1,j}(g_{j}^{m}t) = R_{1,j}(s) = R_{1,j}(g_{j}^{l}t) if and only if l = m, i.e., β = j. □

Theorem 4.5. Let μ = (a, 1^{b}) with a + b = dp and b = sp + b_{0} with 0 ≤ b_{0} ≤ p − 1. If b_{0} is even, then the stable generic Jordan type of S^{μ}↓_{E_{d}} is (1^{\binom{d−1}{s}}); otherwise, it

is ((p − 1)^{\binom{d−1}{s}}).

Proof. We prove the result by induction on b. If b = 0, then S^{μ}↓_{E_{d}} is the trivial kE_{d}-module. So it has Jordan type (1). Suppose that we know the stable generic Jordan type of S^{μ}↓_{E_{d}} for some μ = (a, 1^{b}) with b = sp + b_{0} and 0 ≤ b_{0} ≤ p − 1. Let λ = (a − 1, 1^{b+1}). Consider two cases, b_{0} < p − 1 or b_{0} = p − 1. In the case b_{0} < p − 1, the p-residue of b + 1 is b_{0} + 1. The kE_{d}-module

S^{(a−1,1^{b})}↑_{𝔖_{dp−1}}^{𝔖_{dp}}↓_{E_{d}}

is generically free. On the other hand, using the Branching Theorem, the module has a filtration with factors S^{μ}↓_{E_{d}}, S^{(a−1,2,1^{b−1})}↓_{E_{d}} and S^{λ}↓_{E_{d}}. Since the p-residue of b is strictly less than p − 1, the partition (a − 1, 2, 1^{b−1}) has non-empty p-core. So S^{(a−1,2,1^{b−1})}↓_{E_{d}} is a direct summand of S^{(a−1,1^{b})}↑_{𝔖_{dp−1}}^{𝔖_{dp}}↓_{E_{d}} and it is generically free. So the stable generic Jordan types of S^{μ}↓_{E_{d}} and S^{λ}↓_{E_{d}} are complementary. In the case b_{0} = p − 1, we have b + 1 ≡ 0(mod p). By the Branching Theorem, S^{(a,1^{b+1})}↓_{E_{d}} has a filtration S^{μ}↓_{E_{d}} and S^{λ}↓_{E_{d}} reading from the top. Let α ∈ k^{d} be a generic point, we construct the short exact sequence

\xymatrix{0 \ar[r] & S^{λ}↓_{⟨u_{α}⟩} \ar[r]^{f} & S^{(a,1^{b+1})}↓_{⟨u_{α}⟩} \ar[r] & S^{μ}↓_{⟨u_{α}⟩} \ar[r] & 0}

where f maps each standard λ-polytabloid e_{t} to the standard (a, 1^{b+1})-polytabloid e_{ϕ(t)} where ϕ(t)_{ij} = t_{ij} if (i, j) ≠ (1, a) and ϕ(t)_{1a} = dp + 1. Note that the set of standard tableaux of (a, 1^{b+1}) is the union of Ω_{1} and Ω_{2} with Ω_{1} ∩ Ω_{2} = ∅ and where Ω_{1} is the set consisting of standard tableaux s such that s_{1a} = dp + 1 and Ω_{2} is the set consisting of standard tableaux s such that s_{b+2,1} = dp + 1. We claim that f splits in the stable k⟨u_{α}⟩-module category. Once we have done this, we have S^{(a,1^{b+1})}↓_{⟨u_{α}⟩} ≅ S^{μ}↓_{⟨u_{α}⟩}⊕S^{λ}↓_{⟨u_{α}⟩}. Using Theorem 4.1 with r = 1 and b+1 = (s+1)p, the stable generic Jordan type of S^{(a,1^{b+1})}↓_{E_{d}} is (1^{\binom{d}{s+1}}). By induction hypothesis,

the stable generic Jordan type of S^{μ}↓_{E_{d}} is (1^{\binom{d−1}{s}}). So the stable generic Jordan type of S^{λ}↓_{E_{d}} is

(1^{\binom{d}{s+1}−\binom{d−1}{s}}) = (1^{\binom{d−1}{s+1}}) .

This completes the inductive step.

We want to define a map g : S^{(a,1^{b+1})}↓_{⟨u_{α}⟩} → S^{λ}↓_{⟨u_{α}⟩} in the stable module category such that gf = id_{S^{λ}↓_{⟨uα⟩}}. For any s ∈ Ω_{1}, we define g(e_{s}) = e_{t} where t is the unique standard μ-tableau such that ϕ(t) = s. Let s ∈ Ω_{2}. If (u_{α} − 1)^{p−1}e_{s} = 0, by Lemma 4.4 with s = d (we have abused the notation s), for each 2 ≤ j ≤ d we have I_{j} ⊆ R_{1}(s) or I_{j} ⊆ C_{1}(s). Since b_{0} = p − 1 and s_{b+2,1} = dp + 1, it follows that I_{1} ⊆ C_{1}(s). So (u_{α} − 1)e_{s} = 0 if (u_{α} − 1)^{p−1}e_{s} = 0. Since e_{s} is fixed by u_{α}, we may define g(e_{s}) = 0. If (u_{α} − 1)^{p−1}e_{s} ≠ 0, we claim that the set {e_{s}, (u_{α} − 1)e_{s}, …, (u_{α} − 1)^{p−1}e_{s}} is k-linearly independent. Suppose that we have a relation

a_{0}e_{s} + a_{1}(u_{α} − 1)e_{s} + … + a_{p−1}(u_{α} − 1)^{p−1}e_{s} = 0

with a_{0}, …, a_{p−1} ∈ k. Note that (u_{α} − 1)^{p} = 0. Multiplying the equation by (u_{α} − 1)^{p−1}, we get a_{0}(u_{α} − 1)^{p−1}e_{s} = 0. Since (u_{α} − 1)^{p−1}e_{s} ≠ 0, we have a_{0} = 0. Inductively, by multiplying (u_{α} − 1)^{i} for some suitable i to the equation, we show that a_{0} = a_{1} = … = a_{p−1} = 0. So e_{s} lies inside a free summand of S^{(a,1^{b+1})}↓_{⟨u_{α}⟩}. In this case, we may define g(e_{s}) = 0. The map g gives a splitting for f in the stable k⟨u_{α}⟩-module category. The proof is complete. □

Combining Corollary 4.2 and Theorem 4.5, we have our main result Theorem 1.3.

Acknowledgement. I would like to thank my supervisor Dave Benson for valuable discussions and Johannes Orlob for pointing out the obvious generalization of 3.2.2 [12] which leads to Theorem 3.1 (ii).

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