THE VARIETIES FOR SOME SPECHT MODULES

KAY JIN LIM

Abstract. J. Carlson introduced the cohomological and rank variety for a module over a finite group algebra. We give a general form for the largest component of the variety for the Specht module for the partition (p^{p}) of p^{2} restricted to a maximal elementary abelian p-subgroup of rank p. We determine the varieties of a large class of Specht modules corresponding to p-regular partitions. To any partition μ of np of not more than p parts with empty p-core we associate a unique partition Φ(μ) of np, where the rank variety of the restricted Specht module S^{μ}↓_{E_{n}} to a maximal elementary abelian p-subgroup E_{n} of rank n is V_{E_{n}}^{♯}(k) if and only if V_{E_{n}}^{♯}(S^{Φ(μ)}) = V_{E_{n}}^{♯}(k). In some cases where Φ(μ) is a 2-part partition, we show that the rank variety V_{E_{n}}^{♯}(S^{μ}) is V_{E_{n}}^{♯}(k). In particular, the complexity of the Specht module S^{μ} is n.

1. Introduction

Over last forty years, group cohomology has been studied extensively, especially its interaction with modular representation theory. Carlson [3] introduced the varieties for modules over group algebras. Various results have been published which relate the properties of the algebraic variety of a module and the structure of the module itself.

Hemmer and Nakano studied the support varieties for permutation modules and Young modules [8] over the symmetric groups. The varieties for most Specht modules remain unknown. A partition is made up of (p × p)-blocks if each part is a multiple of p and each part is repeated a multiple of p times. The VIGRE research group in Georgia (2004) conjectured that the variety for the Specht module corresponding to a partition μ is the variety of the defect group of the block in which the Specht module lies unless and only unless the partition is made up of (p × p)-blocks. According to the referee, the VIGRE research group proved one direction of the conjecture. Namely, if a partition μ is made up of (p × p)-blocks, then the variety for the Specht module S^{μ} is never the variety of the defect group of the block containing S^{μ} (see also [7]).

It is not difficult to verify the conjecture when μ has p-weight strictly less than p. In the first part of this paper, we study the variety for the Specht module corresponding to the partition (p^{p}). In the latter part, we verify the conjecture for a large class of partitions of p^{2} and np for some positive integer n, where they are not made up of (p × p)-blocks.

We organize this paper as follows. A brief introduction to the varieties for modules and representation theory of symmetric groups is given in Section 2, the standard texts are [2] II, [10] and [12]. We also set up notations in this section. Our main results are stated in Section 3. In Section 4, in the case p is odd, we show that the dimension of the rank variety for the Specht module S^{(p^{p})} is p − 1 . This therefore gives the complexity of the Specht module. In [4], Carlson gives an upper bound for the degree of the projectivized rank variety for a module over an elementary abelian p-group. Later in this section, we show that the radical ideal corresponding to the largest component of S^{(p^{p})}↓_{E_{p}} is generated by a single non-zero polynomial, where a general formula is given. This general polynomial shows that the degree of the projectivized rank variety for the module S^{(p^{p})}↓_{E} is non-zero and divisible by (p − 1)^{2}.

In Section 5, we determine the varieties of a large class of Specht modules corresponding to p-regular partitions. For each partition μ of np with no more than p parts which has empty p-core, we associate a unique partition Φ(μ) of np such that each part of the partition is a multiple of p and with the property that the rank variety V_{E_{n}}^{♯}(S^{μ}) is V_{E_{n}}^{♯}(k) if and only if V_{E_{n}}^{♯}(S^{Φ(μ)}) = V_{E_{n}}^{♯}(k). In some cases where Φ(μ) is a 2-part partition, we show that the rank variety for the Specht module S^{μ} restricted to a maximal elementary abelian p-subgroup E of the symmetric group 𝔖_{np} of rank n is precisely the rank variety for the trivial module. In this case, the complexity of the Specht module is n. At the end of this section, we give a complete list of the varieties for the modules S^{μ}↓_{E_{3}} where μ are partitions of 9, over p = 3.

2. Background Materials and Notations

Throughout this paper p is a prime and k is an algebraically closed field of characteristic p. Let G be a finite group whose order is divisible by p and M be a finitely generated kG-module. Carlson [3] introduced the cohomological variety V_{G}(M) for the module M. Avrunin and Scott show that the variety V_{G}(M) has a stratification [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 map res_{G,E} : Ext_{kG}^{•}(k, k) → Ext_{kE}^{•}(k, k). The complexity of the module M is precisely the dimension dim V_{G}(M) of the variety V_{G}(M). Since the map res_{G,E}^{∗} is a finite map, we have

dim V_{G}(M) = max_{E∈E(G)}{dim res_{G,E}^{∗} V_{E}(M)} = max_{E∈E(G)}{dim V_{E}(M)}

Let E be an elementary abelian group of order p^{n} with generators g_{1}, …, g_{n} and M be a finitely generated kE-module. For a generic affine point 0 ≠ α = (α_{1}, …, α_{n}) ∈ k^{n}, the Jordan type of the restriction M↓_{k⟨u_{α}⟩} is called the generic Jordan type of the module M, where u_{α} = 1 + ∑_{i=1}^{n} α_{i}(g_{i} − 1) (see [3] and [6]). Generic Jordan type is compatible with direct sum and tensor product of two kE-modules, where the tensor product is given by the diagonal E-action (Proposition 4.7 [6]). The stable generic Jordan type of M is the generic Jordan type of M modulo projective direct summands. The rank variety V_{E}^{♯}(M) of the kE-module M is the set

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

It is isomorphic to the cohomological variety of the module M (via the Frobenius map when p is odd) [1]. Note that M is not generically free if and only if its rank variety V_{E}^{♯}(M) is V_{E}^{♯}(k).

Proposition 2.1. Let M, N be kG-modules.

(i) V_{G}(M) = {0} if and only if M is projective kG-module.

(ii) V_{G}(M ⊕ N) = V_{G}(M) ∪ V_{G}(N) and V_{G}(M ⊗_{k} N) = V_{G}(M) ∩ V_{G}(N) with the

diagonal G-action. (iii) If p |̸ dim_{k} M, then V_{G}(M) = V_{G}(k). (iv) Suppose that M is an indecomposable kG-module in a block with defect group

D. Then V_{G}(M) = res_{G,D}^{∗} V_{D}(M). Let |G| = p^{a}s with gcd(s, p) = 1. Then p^{a}/|D| divides dim_{k} M. Furthermore, if

gcd (\frac{|D| dim_{k} M}{p^{a}} , p) = 1

then V_{G}(M) = res_{G,D}^{∗} V_{D}(k).

(v) If H be a subgroup of G, then V_{H}(M) = (res_{G,H}^{∗})^{−1} V_{G}(M).

(vi) The variety V_{G}(M) is a homogeneous subvariety of V_{G}(k). If M is indecomposable, then the projectivized variety \overline{V_{G}(M)} is connected when endowed with the Zariski topology.

(vii) Suppose further that G is an elementary abelian p-group of rank n and r = dim V_{G}(M). Then p^{n−r} divides dim_{k} M.

The standard texts for the representation of symmetric groups are [10] and [12]. Let 𝔖_{m} denote the symmetric group on m letters. For each s ≤ n, let E_{s} be the elementary abelian subgroup of 𝔖_{np} generated by the p-cycles ((i − 1)p + 1, …, ip) for 1 ≤ i ≤ s. A partition μ of m is a sequence of positive integers (μ_{1}, μ_{2}, …, μ_{s}) such that μ_{1} ≥ μ_{2} ≥ … ≥ μ_{s} and ∑_{i=1}^{s} μ_{i} = m; in this case we write |μ| = m. The Young subgroup 𝔖_{μ} of 𝔖_{m} 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 permutation module M^{μ} ≅ k_{𝔖_{μ}}↑^{𝔖_{m}} is the k𝔖_{m}-module k-spanned by all μtabloids {t} where 𝔖_{m} acts by permuting the numbers assigned to the nodes of t. The Specht module S^{μ} is the submodule of M^{μ} k-spanned by the μ-polytabloids e_{t} = ∑_{σ∈C_{t}} sgn(σ)σ{t} where C_{t} is the column stabilizer of the μ-tableau t. It has dimension given by the hook formula

dim_{k} S^{μ} = \frac{m!}{∏_{i,j∈[μ]} h_{ij}}

where h_{ij} is the hook length of the (i, j)-node of the Young diagram [μ] of μ. The number of skew p-hooks removed to obtain the p-core μ̃ of μ is precisely the number of hook lengths of μ divisible by p, we write m_{μ} for this common number. A defect group D_{μ} of the block containing the Specht module S^{μ} is a Sylow p-subgroup of the symmetric group 𝔖_{m_{μ}p}.

The Young module Y^{μ} corresponding to the partition μ is an indecomposable direct summand of M^{μ} with the property that each indecomposable summand of M^{μ} is isomorphic to some Y^{λ} with λ ⊵ μ by the lexicographic ordering (Theorem 3.1 of [11]) and Y^{μ} occurs with multiplicity one. Each Y^{μ} has a Specht filtration and the filtration multiplicity is well-defined [5]. The permutation module M^{(r−k,k)} corresponding to a two part partition (r − k, k) has a Specht filtration with factors

S^{(r)}, S^{(r−1,1)}, …, S^{(r−k,k)}

reading from the top (Theorem 17.13 of [10]). So any Young module Y^{λ} appearing in the decomposition of M^{(r−k,k)} satisfies λ = (r − s, s) for some s ≤ k and has multiplicity one.

Proposition 2.2.

(i) [The Branching Theorem (§9 of [10])] Let μ be a partition of m and Ω(μ) be the set consisting of all partitions obtained by removing a node from μ, then the k𝔖_{m−1}-module S^{μ}↓_{𝔖_{m−1}} has a filtration with Specht factors S^{λ} one for each λ ∈ Ω(μ).

(ii) [Nakayama’s Conjecture] Let μ, λ be partitions of m. Then the Specht modules S^{μ}, S^{λ} lie in the same block if and only if μ̃ = λ̃. So any k𝔖_{m}-module which has a filtration with Specht factors decomposes into direct summands according to p-cores.

(iii) Let C_{p} be the cyclic group of order p and J_{i} be the Jordan block of size 0 ≤ i ≤ p. Then there is an extension of indecomposable kC_{p}-modules

0 → J_{i} → J_{a} ⊕ J_{b} → J_{j} → 0

with a ≥ b if and only if max{i, j} ≤ a ≤ p, 0 ≤ b ≤ min{i, j} and a + b = i + j.

(iv) If μ is a partition of np with non-empty p-core, then S^{μ}↓_{E_{n}} is generically

free.

(v) Let μ′ be the conjugate of a partition μ of m. Then V_{𝔖_{m}}(S^{μ′}) = V_{𝔖_{m}}(S^{μ}). In particular, V_{E}^{♯}(S^{μ′}) = V_{E}^{♯}(S^{μ}) for any elementary abelian p-subgroup E of 𝔖_{m}.

Proof. (iv) By Proposition 2.1 (iv), V_{𝔖_{np}}(S^{μ}) = res_{𝔖_{np},D_{μ}}^{∗} V_{D_{μ}}(S^{μ}) where D_{μ} is a Sylow p-subgroup of 𝔖_{m_{μ}p}. Since m_{μ} < n, any maximal elementary abelian psubgroup of D_{μ} ≤ 𝔖_{m_{μ}p} has rank strictly less than n. So

dim V_{E_{n}}(S^{μ}) ≤ dim V_{𝔖_{np}}(S^{μ}) = dim V_{D_{μ}}(S^{μ}) ≤ max_{E≤D_{μ}}{dim V_{E}(S^{μ})} < n

where E runs through all elementary abelian p-subgroups of D_{μ} by the Quillen Stratification Theorem.

(v) Note that S^{μ′} ⊗_{k} S^{(1^{m})} ≅ (S^{μ})^{∗} (see 8.15 [10]). By 5.7.3 and 5.8.5 of [2] II, we have V_{𝔖_{m}}(S^{μ′}) = V_{𝔖_{m}}(S^{μ′}) ∩ V_{𝔖_{m}}(k) = V_{𝔖_{m}} ((S^{μ})^{∗}) = V_{𝔖_{m}}(S^{μ}). □

3. Main Results

The VIGRE research group in Georgia (2004) conjectured that the variety for the Specht module corresponding to a partition μ is the variety of the defect group where the Specht module lies in if and only if the partition is not made up of (p×p)- blocks. According to the referee, the VIGRE research group proved one direction of the conjecture. Namely, if a partition μ is made up of (p × p)-blocks, then the variety for the Specht module S^{μ} is never the variety of the defect group of the block containing S^{μ}. Let W_{i}(S^{(p^{p})}) be the union of all irreducible components of V_{E_{p}}^{♯}(S^{(p^{p})}) ⊆ V_{E_{p}}^{♯}(k) of dimension i. In Section 4, we prove the following theorem.

Theorem 3.1. Suppose that p is an odd prime and μ = (p^{p}) is the (p×p)-partition. (i) dim V_{𝔖_{p^{2}}}(S^{μ}) = p − 1. (ii) We have I(W_{p−1}(S^{(p^{p})})) = \sqrt{f} = ⟨f⟩ where

f(x_{1}, …, x_{p}) = (x_{1} … x_{p})^{p−1}f̃+ ∑_{i=1}^{p} x_{1}^{n(p−1)} … \begin{matrix} \\ x_{i} & n(p−1) \end{matrix} … x_{p}^{n(p−1)}

for some homogeneous polynomial f̃ ∈ k[x_{1}, …, x_{p}]^{(𝔽_{p}^{×})^{p}⋊𝔖_{p}} and positive in-

teger n where x_{1}^{n(p−1)} … \begin{matrix} \\ x_{i} & n(p−1) \end{matrix} … x_{p}^{n(p−1)} is the product of all x_{j}^{n(p−1)}’s,

1 ≤ j ≤ p, except the term x_{i}^{n(p−1)}. In Section 5, we determine the varieties of a large class of Specht modules that

are not made up of (p × p)-blocks. We associate to any partition μ = (μ_{1}, …, μ_{s}) of np not more than p parts with empty p-core a unique partition Φ(μ) of np such that Φ(μ) = (n_{1}p, n_{2}p, …, n_{r}p) and r ≤ s. Hypothesis 3.2. Suppose that μ is a partition of np not more than p parts with empty p-core and one of the following conditions hold. (H1) The prime p is odd, n = p and Φ(μ) is a 2-part partition (p^{2} − mp, mp) of

p^{2} for some 1 ≤ m < p/2. (H2) The prime p is odd and Φ(μ) is a 2-part partition (np − εp, εp) such that

n ≢ 2(mod p) and ε ∈ {1, 2}. (H3) Φ(μ) = (np). (H4) p = 2 and Φ(μ) is the partition (2n − 2, 2) ≠ (2, 2) or (2n − 4, 4) ≠ (4, 4).

Theorem 3.3. If μ is a partition satisfying Hypothesis 3.2, then V_{E_{n}}^{♯}(S^{μ}) = V_{E_{n}}^{♯}(k). In this case, the complexity of S^{μ} is n.

4. The Variety for the Specht Module S^{(p^{p})}

Theorem 4.1. Suppose that the Specht module S^{μ} corresponding to a partition μ of n lies inside a block with abelian defect D_{μ}. We have V_{𝔖_{n}}(S^{μ}) = res_{𝔖_{n},D_{μ}}^{∗} V_{D_{μ}}(k) and the complexity of the Specht module S^{μ} is exactly the p-weight of μ. Proof. Let n! = p^{a}b with gcd(a, b) = 1 and m_{μ} be the p-weight of μ. Since the number of hook lengths of [μ] divisible by p is at least as many as m_{μ}, we have

ℤ ∋ \frac{|D_{μ}|}{p^{a}}(dim_{k} S^{μ})_{p} = \frac{p^{m_{μ}}}{p^{a}} \frac{p^{a}}{∏_{(i,j)∈[μ]}(h_{ij})_{p}} = 1

By Proposition 2.1 (iv), V_{𝔖_{n}}(S^{μ}) = res_{𝔖_{n},D_{μ}}^{∗} V_{D_{μ}}(k). The complexity of the S^{μ} is exactly the p-rank of the defect group D_{μ}, i.e., m_{μ}. □

For p = 2, let F be the maximal abelian subgroup of 𝔖_{4} generated by (12)(34) and (13)(24). The module S^{(2^{2})}↓_{F} is isomorphic to k ⊕ k, so dim V_{𝔖_{4}}(S^{(2^{2})}) = 2. For the rest of Section 4, p is an odd prime.

4.1. Some vanishing ideals. Given the symmetric group 𝔖_{m} of degree m, the group (𝔽_{p}^{×})^{m} ⋊ 𝔖_{m} is defined by the group actions (β, σ)(β′, σ′) = (β · σ(β′), σσ′) where σ(β′) = (β′_{σ^{−1}(1)}, …, β′_{σ^{−1}(m)}) for all σ, σ′ ∈ 𝔖_{m} and β = (β_{1}, …, β_{m}), β′ = (β′_{1}, …, β′_{m}) ∈ (𝔽_{p}^{×})^{m}. This group acts on the polynomial ring k[x_{1}, …, x_{m}] via

(β, σ)x_{i} := β_{σ(i)}x_{σ(i)}

for all 1 ≤ i ≤ m and (β, σ) ∈ (𝔽_{p}^{×})^{m} ⋊ 𝔖_{m}. We think of the action in two stages, first by (𝔽_{p}^{×})^{m} and then followed by 𝔖_{m}.

Lemma 4.2. Let p be an odd prime, m ≥ 3, f ∈ k[x_{1}, …, x_{m}]^{(𝔽_{p}^{×})^{m}⋊A_{m}} and x_{i} be a factor of f for some 1 ≤ i ≤ m. Then x_{1}^{p−1} … x_{m}^{p−1} is a factor of f.

Proof. Since f ∈ (k[x_{1}, …, x_{m}]^{(𝔽_{p}^{×})^{m}})^{A_{m}} = k[x_{1}^{p−1}, …, x_{m}^{p−1}]^{A_{m}}, it follows that if x_{i} divides f then x_{i}^{p−1} divides f. As f is invariant under the action of A_{m}, all x_{j}^{p−1}’s divide f. All factors x_{j}^{p−1} are pairwise coprime, so the product divides f. □

Lemma 4.3. Suppose that m ≥ 3, G = (𝔽_{p}^{×})^{m} ⋊𝔖_{m} and f is a non-zero polynomial in k[x_{1}, …, x_{m}] such that the ideal ⟨f⟩ generated by f satisfies the property G⟨f⟩ = ⟨f⟩, i.e., for all g ∈ G, we have gf ∈ ⟨f⟩. Then f is either fixed by the subgroup (𝔽_{p}^{×})^{m} ⋊ A_{m} of G or divisible by x_{1} … x_{m}. In the first of these cases, f is either fixed or negated by the action of G.

Proof. Write R = k[x_{1}, …, x_{m}] and f = ∑ a_{n_{1}...n_{m}}x_{1}^{n_{1}} … x_{m}^{n_{m}} with a_{n_{1}...n_{m}} ∈ k. Suppose that for each (β, σ) ∈ G, we have (β, σ)f = hf for some polynomial h ∈ R. Since the action of (β, σ) on each monomial of f is multiplication by some β_{i}’s and permuting x_{j}’s, the highest degree of a monomial appearing in f is at least as large as the highest degree of a monomial appearing in hf. So, h = c(β, σ) lies in k. Let 1 = (1, …, 1) in (𝔽_{p}^{×})^{m}. Let β(i) be the element in (𝔽_{p}^{×})^{m} such that its ith coordinate is β_{i} and 1 elsewhere. It is clear that c(1, σ)^{−1} = c(1, σ^{−1}) and

c(β, σ) = c(β(1), 1)c(β(2), 1) … c(β(m), 1)c(1, σ)

for all β ∈ (𝔽_{p}^{×})^{m} and σ ∈ 𝔖_{m}.

It suffices to examine the action of transpositions on f to determine c(1, σ). For any transposition σ, f = σ^{2}f = c(1, σ)^{2}f, so c(1, σ) ∈ {±1}. This shows that c(1, ξ) = 1 for all ξ ∈ A_{m}. Suppose that there exists some monomial

a_{n_{1}...n_{m}}x_{1}^{n_{1}} … x_{m}^{n_{m}}

involved in f with n_{i} = 0 for some 1 ≤ i ≤ m. By the action of β(i) on f we have a_{n_{1}...n_{m}} = c(β(i), 1)a_{n_{1}...n_{m}}, i.e., c(β(i), 1) = 1. For any 1 ≤ j ≤ m, let σ ∈ A_{m} such that τ(i) = j, then τf = f gives c(β(j), 1) = 1. So (𝔽_{p}^{×})^{m} ⋊ A_{m} acts trivially on f in this case. Otherwise, every monomial involved in f is divisible by x_{1} and so f is divisible by x_{1} … x_{m}. □

For any fixed 1 ≤ i ≤ m, define x_{1} … x̂_{i} … x_{m} as the product of all x_{j}’s where 1 ≤ j ≠ i ≤ m. We write I_{i} for the radical ideal of the polynomial ring k[x_{1}, …, x_{m}] generated by x_{1} … x̂_{i} … x_{m} and x_{i} .

Lemma 4.4. For m ≥ 2, the vanishing ideal of the union of the varieties V (I_{i}) where 1 ≤ i ≤ m is ({x_{1} … x̂_{i} … x_{m} | 1 ≤ i ≤ m}).

Proof. The algebraic variety given by the union of all V (I_{i})’s consists of all planes at least two coordinates having value 0, which is precisely the algebraic variety defined by the ideal J generated by x_{1} … x̂_{i} … x_{m} for all 1 ≤ i ≤ m. Furthermore, J is a radical ideal, so both sides coincide using Hilbert’s Nullstellensatz. □

Proposition 4.5. Let m ≥ 2 and I be a radical ideal of k[x_{1}, …, x_{m}] generated by a single polynomial f such that V (f) ∩ V (x_{i}) = V (x_{1} … x̂_{i} … x_{m}, x_{i}) for any 1 ≤ i ≤ m. Then

f(x_{1}, …, x_{m}) = x_{1} … x_{m}f̃+ ∑_{i=1}^{m} a_{i}x_{1}^{n_{1i}} … x_{m}^{n_{mi}}

such that f̃ ∈ k[x_{1}, …, x_{m}] and (i) n_{ii} = 0 for all 1 ≤ i ≤ m, (ii) 0 ≠ a_{i} ∈ k for all 1 ≤ i ≤ m, (iii) n_{ij} > 0 for all i ≠ j.

Proof. The given hypothesis implies ⋃ V (x_{1} … x̂_{i} … x_{m}, x_{i}) ⊆ V (f). By Lemma 4.4, f lies inside the ideal ({x_{1} … x̂_{i} … x_{m} | 1 ≤ i ≤ m}), i.e.,

f(x_{1}, …, x_{m}) = ∑_{i=1}^{m} f_{i}x_{1} … x̂_{i} … x_{m}

for some f_{1}, …, f_{m} ∈ k[x_{1}, …, x_{m}]. It is not difficult to see that we can rewrite f as follows

f(x_{1}, …, x_{m}) = x_{1} … x_{m}f̃+ ∑_{i=1}^{m} a_{i}(x_{1}, …, x_{m})x_{1}^{n_{1i}} … x_{m}^{n_{mi}}  (∗)

such that (i) n_{ij} > 0 if j ≠ i and n_{ii} = 0 for all 1 ≤ i ≤ m, (ii) f̃, a_{i} ∈ k[x_{1}, …, x_{m}] for each 1 ≤ i ≤ m, (iii) for all 1 ≤ i, j ≤ m, x_{j} does not divide a_{i} and a_{i} is independent of the

variable x_{i}. For each 1 ≤ i ≤ m, the equation f(x_{1}, …, x_{m}) |_{x_{i}=0} = a_{i}x_{1}^{n_{1i}} … x_{m}^{n_{mi}} implies

V (a_{i}, x_{i}) = V (a_{i}) ∩ V (x_{i}) ⊆ V (f) ∩ V (x_{i}) = V (x_{1} … x̂_{i} … x_{m}, x_{i})

By Hilbert’s Nullstellensatz, we have (x_{1} … x̂_{i} … x_{m}, x_{i}) ⊆ (a_{i}, x_{i}). Suppose that x_{1} … x̂_{i} … x_{m} = a_{i}w_{i} + x_{i}v_{i} for some w_{i}, v_{i} ∈ k[x_{1}, …, x_{m}]. We write w_{i} = u_{i} + x_{i}u′_{i} such that u_{i} is independent of the variable x_{i}. The equation

x_{1} … x̂_{i} … x_{m} − a_{i}u_{i} = a_{i}x_{i}u′_{i} + x_{i}v_{i}

implies that a_{i}x_{i}u′_{i} + x_{i}v_{i} = 0 as the left-hand side is independent of x_{i}. So we have x_{1} … x̂_{i} … x_{m} = a_{i}u_{i}.

Since x_{j} does not divide a_{i} for all j ≠ i, we have u_{i}/x_{1} … x̂_{i} … x_{m} is a polynomial. Comparing degrees, we have a_{i} ∈ k. Note that a_{i} ≠ 0; otherwise,

(x_{1} … x̂_{i} … x_{m}, x_{i}) ⊆ (x_{i})

Now we have all the desired properties. □

4.2. Proof of Theorem 3.1. Let E = E_{p}. Now we are ready to prove Theorem 3.1 (i).

Proof of Theorem 3.1(i). Recall that μ = (p^{p}). By the Branching Theorem, we have S^{μ}↓_{𝔖_{p^{2}−1}} ≅ S^{τ} where τ = (p^{p−1}, p − 1). Note that τ has p-core τ̃ = (p, 1^{p−1}) and the corresponding block has defect group D_{τ} = E_{p−2} ≤ 𝔖_{p^{2}−1}. Using the hook formula

\frac{|D_{τ}|(dim_{k} S^{τ})_{p}}{|𝔖_{p^{2}−1}|_{p}} = \frac{p^{p−2}p}{p^{p−1}} = 1

so V_{𝔖_{p^{2}−1}}(S^{μ}) = V_{𝔖_{p^{2}−1}}(S^{τ}) = res_{𝔖_{p^{2}−1},E_{p−2}}^{∗} V_{E_{p−2}}(k) by Proposition 2.1 (iv). This also shows that dim V_{𝔖_{p^{2}−1}}(S^{μ}) = p−2 and so dim V_{𝔖_{p^{2}}}(S^{μ}) ≤ p−1: if dim V_{𝔖_{p^{2}}}(S^{μ}) = p, then V_{E}(S^{μ}) = V_{E}(k) and so dim V_{𝔖_{p^{2}−1}}(S^{μ}) = dim V_{E_{p−1}}(S^{μ}) = p − 1. On the other hand, let r = dim V_{E}(S^{μ}) ≤ dim V_{𝔖_{p^{2}}}(S^{μ}), using Proposition 2.1 (vii), p^{p−r} divides (dim_{k} S^{μ})_{p} = p, this implies p − 1 ≤ r. So dim V_{𝔖_{p^{2}}}(S^{μ}) = p − 1. □

The quotient group G = N_{𝔖_{p^{2}}}(E)/C_{𝔖_{p^{2}}}(E) ≅ (𝔽_{p}^{×})^{p} ⋊ 𝔖_{p} acts on k^{p} via

(β, σ)α = (β_{σ(1)}α_{σ(1)}, …, β_{σ(p)}α_{σ(p)})

for all α = (α_{1}, …, α_{p}) ∈ k^{p}. Let J be the Jacobson radical of kE. For each 1 ≤ i ≤ p and b ∈ 𝔽_{p}^{×}, we have g_{i}^{b} − 1 ≡ b(g_{i} − 1)(mod J^{2}). Lemma 6.4 of [3] shows that G acts on the rank variety V_{E}^{♯}(M) of a kE-module M and hence on the vanishing ideal I(V_{E}^{♯}(M)) of V_{E}^{♯}(M) where the action is given by (β, σ)x_{i} = β_{σ(i)}x_{σ(i)}.

Lemma 4.6. Let τ = (p^{p−1}, p − 1) and E_{p−1} ≤ 𝔖_{p^{2}−1}. Then the rank variety V_{E_{p−1}}^{♯}(S^{τ}) ⊆ k^{p−1} is the union of all hyperplanes V (x_{i}) ⊆ k^{p−1} where 1 ≤ i ≤ p − 1.

Proof. We knew that V_{𝔖_{p^{2}−1}}(S^{τ}) = res_{𝔖_{p^{2}−1},E_{p−2}}^{∗} V_{E_{p−2}}(k). Apply Proposition 2.1

(v), we have V_{E_{p−1}}(S^{τ}) = (res_{𝔖_{p^{2}−1},E_{p−1}}^{∗})^{−1} res_{𝔖_{p^{2}−1},E_{p−2}}^{∗} V_{E_{p−2}}(k). So the rank vari-

ety V_{E_{p−1}}^{♯}(S^{τ}) lies in the union of hyperplanes ⋃_{i=1}^{p−1} V (x_{i}). Since the rank variety contains V (x_{p−1}) so it contains V (x_{i})’s where 1 ≤ i ≤ p − 1 given by the action of 𝔖_{p−1} on the p − 1 coordinates of V_{E_{p−1}}^{♯}(S^{τ}). □

Since k[x_{1}, …, x_{m}] is a unique factorization domain, prime ideals of height one are principal. Let W_{i}(S^{(p^{p})}) be the union of all irreducible components of V_{E}^{♯}(S^{(p^{p})}) ⊆ k^{p} of dimension i. Since W_{p−1}(S^{(p^{p})}) has codimension 1, by the previous remark, it is defined by a single polynomial in variables x_{1}, …, x_{p}. By Lemma 4.6,

W_{p−1}(S^{(p^{p})}) ∩ V (x_{i}) ⊆ V_{E}^{♯}(S^{(p^{p})}) ∩ V (x_{i}) = ⋃_{j≠i} V (x_{j}, x_{i})

and dim (W_{p−1}(S^{(p^{p})}) ∩ V (x_{i})) = p − 2, so W_{p−1}(S^{(p^{p})}) ∩ V (x_{i}) contains one of the irreducible varieties V (x_{j}, x_{i}). Since W_{p−1}(S^{(p^{p})}) is invariant under the action of G, its intersection with V (x_{i}) contains all of V (x_{j}, x_{i}) where 1 ≤ j ≠ i ≤ p. We have proved the following lemma.

Lemma 4.7. For any 1 ≤ i ≤ p, we have W_{p−1}(S^{(p^{p})}) ∩ V (x_{i}) = ⋃_{j≠i} V (x_{j}, x_{i}).

Proof of Theorem 3.1(ii). By Lemma 4.7 and Proposition 4.5, f has the form

f(x_{1}, …, x_{p}) = x_{1} … x_{p}f̃′ + ∑_{i=1}^{p} a_{i}x_{1}^{n_{1i}} … x_{p}^{n_{pi}}

such that f̃′ ∈ k[x_{1}, …, x_{p}], n_{ij} > 0 for all j ≠ i, n_{ii} = 0 for all 1 ≤ i ≤ p and a_{i} ∈ k^{×} for all 1 ≤ i ≤ p. The group (𝔽_{p}^{×})^{p} ⋊ 𝔖_{p} acts on the radical ideal generated by f. If f is divisible by x_{1} … x_{p}, then V (f) ∩ V (x_{1}) contains V (x_{1}), which contradicts Lemma 4.7. By Lemma 4.3, f is fixed by (𝔽_{p}^{×})^{p} ⋊ 𝔖_{p} up to a sign and fixed by the subgroup S = (𝔽_{p}^{×})^{p} ⋊ A_{p}.

We write w_{i} for x_{1}^{n_{1i}} … x_{p}^{n_{pi}}. Let τ = (12 … p) ∈ A_{p} be the p-cycle. For any 1 ≤ j ≤ p, we have τ^{j−1}f = f. So

(τ^{j−1}f̃′ − f̃′)x_{1} … x_{p} = (a_{j}w_{j} − a_{1}τ^{j−1}w_{1}) + terms divisible by x_{j}

where both w_{j}, τ^{j−1}w_{1} are independent of x_{j}. We must have a_{j} = a_{1} and w_{j} = τ^{j−1}w_{1}. Let a = a_{1} = … = a_{p} ≠ 0. For any β = (β_{1}, …, β_{p}) ∈ (𝔽_{p}^{×})^{p}, let β(j) be the element in (𝔽_{p}^{×})^{p} such that its jth coordinate is β_{j} and 1 elsewhere. Note that β(j)f = f. So

(β_{j}(β(j)f̃′) − f̃′)x_{1} … x_{p} = ∑_{i≠j}(a − aβ_{j}^{n_{ji}})w_{i}

For i ≠ j, x_{i} divides the left-hand term and all w_{r}’s such that r ≠ i, j on the righthand side. So a − aβ^{n_{ji}} = 0, i.e., β_{j}^{n_{ji}} = 1. This shows that for i ≠ j each n_{ji} is divisible by p − 1.

Note that V (1/af) = V (f), so we may assume that f has the form

f(x_{1}, …, x_{p}) = x_{1} … x_{p}f̃+ ∑_{i=0}^{p−1} τ^{i}(x_{2}^{n_{2}(p−1)} … x_{p}^{n_{p}(p−1)})  (∗)

where w_{j} = τ^{j−1}x_{2}^{n_{2}(p−1)} … x_{p}^{n_{p}(p−1)}. The sum and the term x_{1} … x_{p}f̃′ in (∗) are clearly invariant under the group S. By Lemma 4.2, x_{1} … x_{p}f̃′ = x_{1}^{p−1} … x_{p}^{p−1}f̃ for some homogeneous polynomial f̃fixed by S. For any j ≠ 2, let (2j)f = ϵf where ϵ ∈ {±1}. So

((2j)f̃− ϵf̃)x_{1} … x_{p} = (ϵaw_{j} − a(2j)w_{2}) + terms divisible by x_{j}

gives us w_{j} = (2j)w_{2}. Comparing the power of x_{j+1}, we have n_{2} = n_{j}. So n_{2} = … = n_{p}. □

Let V ⊆ ℙ^{n−1}(k) be a projective variety of dimension m. The intersection of V and a generic linear subspace W of ℙ^{n−1} of dimension n − m − 1 is a union of s points for some fixed positive integer s. This positive integer is the degree of the projective variety V. In case dim W < n − m − 1, then V ∩ W = ∅.

Corollary 4.8. For p ≥ 3, the degree of the projectivized variety \overline{V_{E}^{♯} (S^{(p^{p})})} is nonzero and divisible by (p − 1)^{2}.

Proof. Let W ⊆ k^{p} be a generic linear subspace of dimension 2. For any component W_{i}(S^{(p^{p})}) such that 0 ≤ i ≤ p − 2, we have W_{i}(S^{(p^{p})}) ∩ W = {0}. So

V_{E}^{♯}(S^{(p^{p})}) ∩ W = W_{p−1}(S^{(p^{p})}) ∩ W = s lines passing through the origin

where s is the degree of the homogeneous variety W_{p−1}(S^{(p^{p})}), i.e., the degree of a homogeneous polynomial f defining W_{p−1}(S^{(p^{p})}) such that \sqrt{f} = ⟨f⟩. So s is non-zero and divisible by (p − 1)^{2}. □

5. The Variety for the Specht Module S^{μ} where μ is p-regular

5.1. The permutation modules. Let m, n be two non-negative integers and m = … + m_{1}p + m_{0}, n = … + n_{1}p + n_{0} be their p-adic expansions. The number m is p-contained in n if m_{i} ≤ n_{i} for all i ≥ 0. In this case we write m ⊆_{p} n. Note that m ⊆_{p} n if and only if \binom{n}{m} ≢ 0(mod p). Corollaries 5.2 and 5.3 rely on Theorem 3.3 of [9].

Theorem 5.1 (3.3 of [9]). The Young module Y^{(r−s,s)} is a direct summand of the permutation module M^{(r−m,m)} if and only if m − s ⊆_{p} r − 2s.

Corollary 5.2. For any integer p < m ≤ p^{2}/2, we have a decomposition

M^{(p^{2}−m,m)} ≅ M^{(p^{2}−m+p,m−p)} ⊕ Q

where Q has a filtration with Specht factors

S^{(p^{2}−m+p−1,m−p+1)}, S^{(p^{2}−m+p−2,m−p+2)}, …, S^{(p^{2}−m,m)}

Proof. It suffices to show that if Y^{(p^{2}−s,s)} is a direct summand of M^{(p^{2}−m+p,m−p)}, then it is also a direct summand of M^{(p^{2}−m,m)}. Note that the trivial module k ≅ Y^{(p^{2})} is not a direct summand of M^{(p^{2}−w,w)} for any 0 < w ≤ p^{2}/2. Let 0 < s ≤ m − p and Y^{(p^{2}−s,s)} be a direct summand of M^{(p^{2}−m+p,m−p)}, i.e.,

a_{1}p + a_{0} = m − p − s ⊆_{p} p^{2} − 2s = b_{1}p + b_{0}

where 0 ≤ a_{0}, a_{1}, b_{0}, b_{1} ≤ p − 1. Note that a_{0} + (a_{1} + 1)p is the p-adic expansion of m − s. If a_{1} + 1 > b_{1}, then a_{1} = b_{1}. So (p^{2} − 2s) − (m − p − s) = b_{0} − a_{0} < p. On the other hand, since s ≤ m ≤ p^{2}/2, we have (p^{2}−2s)−(m−p−s) = p^{2}+p−(m+s) ≥ p, which is a contradiction. This shows that m − s ⊆_{p} p^{2} − 2s. □

Corollary 5.3. Let p be an odd prime.

(i) If n ≢ 1, 2(mod p), then M^{(np−2p,2p)} ≅ M^{(np−p,p)} ⊕ Q; (ii) If n ≡ 1(mod p), then M^{(np−p,p)} ≅ k ⊕ N for some k𝔖_{np}-module N and

M^{(np−2p,2p)} ≅ N ⊕ R. In both cases, Q, R have filtrations with Specht factors

S^{(np−p−1,p+1)}, S^{(np−p−2,p+2)}, …, S^{(np−2p,2p)}

Proof. Suppose that 0 < s ≤ p and let Y^{(np−s,s)} be a direct summand of M^{(np−p,p)}, i.e.,

a_{0} = p − s ⊆_{p} np − 2s = … + b_{2}p^{2} + b_{1}p + b_{0}

where 0 ≤ a_{0}, b_{i} < p for all i ≥ 0. Note that b_{0} + 2s ≡ 0(mod p) and a_{0} + p is the p-adic expansion of 2p − s. On the other hand, we have

p + 1 ≤ p + s = a_{0} + 2s ≤ b_{0} + 2s ≤ (p − 1) + 2p = 3p − 1

So b_{0} + 2s = 2p. Substituting into the p-adic expansion of np − 2s, we see that n ≡ 2(mod p) unless b_{1} ≠ 0. This shows that 2p − s ⊆_{p} np − 2s in both (i) and (ii). Suppose that n ≢ 1(mod p), the equation

\binom{np}{2p} = \binom{np}{p} · \frac{(np − 2p + 1)(np − 2p + 2) … (np − p)}{(p + 1)(p + 2) … 2p}

implies that the trivial module k is a direct summand of M^{(np−p,p)} if and only if it is a direct summand of M^{(np−2p,2p)}. So (i) is proved. If n ≡ 1(mod p), then k is a direct summand of M^{(np−p,p)} but not of M^{(np−2p,2p)}. □

Lemma 5.4. Let μ = (n_{1}p, …, n_{s}p) be a partition of np such that n_{1} ≥ n_{2} ≥ … ≥ n_{s} > 0. Then the generic Jordan type of the module M^{μ}↓_{E_{n}} is (p^{a}, 1^{b}) where

a = \frac{dim_{k} M^{μ} − b}{p}, b = \frac{n!}{n_{1}!n_{2}! … n_{s}!}

Proof. The permutation module M^{μ} is isomorphic to the trivial module induced from the Young subgroup 𝔖_{μ} to 𝔖_{np}. Apply the Mackey decomposition formula

M^{μ}↓_{E_{n}} ≅ k_{𝔖_{μ}}↑^{𝔖_{np}}↓_{E_{n}} ≅ ⨁_{E_{n}g𝔖_{μ}} ^{g}k↓_{^{g}𝔖_{μ}∩E_{n}}↑^{E_{n}}

Double coset representatives of E_{n}, 𝔖_{μ} in 𝔖_{np} correspond to the orbits O_{E_{n}}(g_{i}) of the action of E_{n} on the left coset representatives g_{1}, …, g_{m} of 𝔖_{μ} in 𝔖_{np}. The stabilizer Stab_{E_{n}}(g_{i}) of g_{i} consists of precisely the elements e ∈ E_{n} such that eg_{i} ∈ g_{i}𝔖_{μ}. So Stab_{E_{n}}(g_{i}) = E_{n} if and only if O_{E_{n}}(g_{i}) = {g_{i}} if and only if the μ-tabloid corresponding to g_{i} is fixed by E_{n}. For each μ-tabloid {t}, denote by R_{i}(t) the set consisting of integers in the ith row of {t}. Take {t_{0}} as the μ-tabloid such that

R_{i}(t_{0}) = {1 + ∑_{j=0}^{i−1} μ_{j}, 2 + ∑_{j=0}^{i−1} μ_{j}, …, ∑_{j=0}^{i} μ_{j}}

So the μ-tabloid corresponding to g_{i} is {g_{i}t_{0}}. Note that the μ-tabloids {t} fixed by E_{n} are precisely those satisfying the property that for each 1 ≤ i ≤ n there exists some 1 ≤ u ≤ s such that {1 + (i − 1)p, …, ip} ⊆ R_{u}(t). So

k_{𝔖_{μ}}↑^{𝔖_{np}}↓_{E_{n}} ≅ ( ⨁_{\begin{matrix}E_{n}g𝔖_{μ} \\ E_{n}∩^{g}𝔖_{μ}<E_{n} \end{matrix}} k_{E_{n}∩^{g}𝔖_{μ}}↑^{E_{n}} ) ⊕ ( ⨁_{\begin{matrix}E_{n}g𝔖_{μ} \\ E_{n}∩^{g}𝔖_{μ}=E_{n} \end{matrix}} k)

The summand k_{E_{n}∩^{g}𝔖_{μ}}↑^{E_{n}} is generically free if E_{n} ∩ ^{g}𝔖_{μ} < E_{n}; otherwise, it has generic Jordan type (1), and there are precisely \frac{(n_{1}+...+n_{s})!}{n_{1}!...n_{s}!} of them. □

5.2. The map Φ. Every partition can be associated to a set of β-numbers and represented by an abacus (§2.7 of [12]). For the rest of our discussion, whenever we speak of an abacus of μ, we mean the abacus associated to the choice of βnumbers given by the first column hook lengths of μ. So the abacus of the partition μ = (μ_{1}, …, μ_{s}) has beads μ_{i} +(s−i) where 1 ≤ i ≤ s. In the case s ≤ p, the p-core μ̃ of μ is empty if and only if for each 1 ≤ i ≤ s there is a unique 0 ≤ j_{i} ≤ s − 1 such that μ_{i} + (s − i) ≡ j_{i}(mod p). If μ_{i} ≢ 0(mod p) for some 1 ≤ i ≤ s, let 1 ≤ b ≤ s be the number such that μ_{b} ≢ 0(mod p) and μ_{i} ≡ 0(mod p) for all b + 1 ≤ i ≤ s. If we insert the beads μ_{s}, μ_{s−1} + 1, …, μ_{b} + (s − b) successively into the abacus and ignore the position of the bead in each runner, we see the following

•  …  •  ∘  …  ∘  •_{b}

where there are s−b beads • on the left-hand side, more than one ∘ in the middle and the last bead •_{b} corresponds to μ_{b}. Since μ̃ = ∅, there is a unique number 1 ≤ a < b such that μ_{a} + (s − a) ≡ μ_{b} + (s − b) − 1(mod p), i.e., the bead immediately to the left of •_{b}.

Hypothesis 5.5. Suppose that μ = (μ_{1}, …, μ_{s}) is a partition with s ≤ p, μ̃ = ∅ and there are unique numbers 1 ≤ a < b ≤ s with the properties:

(i) μ_{i} ≡ 0(mod p) for all b + 1 ≤ i ≤ s and μ_{b} ≢ 0(mod p), (ii) μ_{a} − a ≡ μ_{b} − 1 − b(mod p).

Let μ be a partition satisfying Hypothesis 5.5, η = (μ_{1}, …, μ_{b−1}, μ_{b}−1, μ_{b+1}, …, μ_{s}) and Ω be the set consisting of all (proper) partitions μ(j) where 1 ≤ j ≤ s + 1 such that μ(j) is the partition obtained from η by adding a node to the end of the jth row (assuming μ_{s+1} = 0). We claim that μ(a), μ(b) ∈ Ω and there is no μ(j) ∈ Ω with empty p-core other than μ = μ(b) and μ(a). Suppose that μ_{a−1} = μ_{a}, we have μ_{a−1} + (s − (a − 1)) ≡ μ_{b} − (s − b)(mod p). This implies that the p-core of μ is not empty. This contradiction shows that μ_{a−1} > μ_{a} and so μ(a) ∈ Ω. In the case j = a, μ(a)_{i} + (s − i) = μ_{i} + (s − i) for all i ≠ a, b and

μ(a)_{a} + (s − a) ≡ μ_{b} + (s − b)(mod p)
μ(a)_{b} + (s − b) ≡ μ_{a} + (s − a)(mod p)

So μ(a) has empty p-core because μ has empty p-core. It is clear that μ = μ(b) ∈ Ω. For 1 ≤ j ≤ s ≤ p such that j ≠ a, b, we have

μ(j)_{b} + (s − b) ≡ μ(j)_{a} + (s − a)(mod p)

So the p-core of μ(j) is not empty. In the case j = s + 1, there are at most p + 1 beads in the abacus of μ(s + 1). It is clear that

μ(s + 1)_{a} + ((s + 1) − a) ≡ μ_{b} + (s − b) ≡ μ(s + 1)_{b} + ((s + 1) − b)(mod p)

So μ(s + 1) has non-empty p-core unless s = p and μ_{b} + (s − b) ≡ 0(mod p), i.e., μ_{b} ≡ b(mod p). In this case, we must have μ_{p} ≡ 0(mod p): if not, then b = p and μ_{b} ≡ 0(mod p), contradicts to the hypothesis of μ. However there are two beads corresponding to μ(p + 1)_{p+1} and μ(p + 1)_{p} lying in the first runner of the abacus, so μ(p + 1) has non-empty p-core.

Fix a positive integer n. Let Λ be the set consisting of all partitions μ = (μ_{1}, …, μ_{s}) of np with s ≤ p and μ̃ = ∅. We define a map ϕ : Λ → Λ as follows.

ϕ(μ) = \{ \begin{matrix}μ(a) & if μ satisfies Hypothesis 5.5 \\ μ & otherwise \end{matrix}

Example 5.6. Let p be an odd prime, μ = (u, v, 2^{m}) with 0 ≤ m ≤ p−2 and μ̃ = ∅; for instance, taking u ≡ p−m−1(mod p) and v ≡ p−m+1(mod p). If m > 0, then b = m + 2 and a = 2. So ϕ(μ) = (u, v + 1, 2^{m−1}, 1). Now we follow the procedure for ϕ(μ), we get b = m + 2 and a = 1. So ϕ^{2}(μ) = (u + 1, v + 1, 2^{m−1}). By induction on the integer m, we see that ϕ^{2m}(μ) = (U, V) with U ≡ p − 1(mod p), V ≡ 1(mod p) and U + V = |μ|. So ϕ^{2m+1}(μ) = (U + 1, V − 1), and indeed ϕ^{t}(μ) = (U + 1, V − 1) for all t ≥ 2m + 1 by our definition.

Definition 5.7. Let μ be a partition not more than p parts with empty p-core. There is a positive integer t(μ) such that for all integers t ≥ t(μ) we have ϕ^{t}(μ) = ϕ^{t(μ)}(μ). We define Φ(μ) = ϕ^{t(μ)}(μ).

Recall from Section 2 the definition of generic Jordan type and stable generic Jordan type of a finitely generated kE-module M where E is an elementary abelian p-group. We have V_{E_{n}}^{♯}(M) = V_{E_{n}}^{♯}(k) if and only if M is not generically free. For each 1 ≤ i ≤ p, we define n_{M}(i) as the number of Jordan blocks of size i in the generic Jordan type of M. In the case M = S^{μ}↓_{E_{n}} where μ is a partition of np, we write n_{μ}(i) for n_{S^{μ}↓_{En}}(i). Let N be another finitely generated kE-module. If n_{M}(i) = n_{N}(p − i) for all 1 ≤ i ≤ p − 1, we say that M, N is a pair of modules of complementary stable Jordan type.

Lemma 5.8. Suppose that μ = (μ_{1}, …, μ_{s}) is a partition of np with 1 ≤ s ≤ p and μ̃ = ∅. Let 1 ≤ a < b ≤ s be the unique numbers such that

(i) μ_{i} ≡ 0(mod p) for all b + 1 ≤ i ≤ s and μ_{b} ≢ 0(mod p), (ii) μ_{a} − a ≡ μ_{b} − 1 − b(mod p).

Then ϕ(μ) has empty p-core and S^{μ}↓_{E_{n}}, S^{ϕ(μ)}↓_{E_{n}} is a pair of modules of complementary stable Jordan type. In particular, V_{E_{n}}^{♯}(S^{μ}) = V_{E_{n}}^{♯}(k) if and only if V_{E_{n}}^{♯} (S^{Φ(μ)}) = V_{E_{n}}^{♯}(k).

Proof. Let η and Ω as before. Since there are precisely two partitions μ, ϕ(μ) in Ω with empty p-core, using Proposition 2.2 (i), (ii) and (iv), we get

S^{η}↑^{𝔖_{np}}↓_{E_{n}} ≅ Q ⊕ generically free direct summand

where Q is a direct summand with factors S^{μ}↓_{E_{n}} and S^{ϕ(μ)}↓_{E_{n}}, and Specht modules not in the principal block contribute to the generically free summand. Since

dim V_{E_{n}}(S^{η}↑^{𝔖_{np}}) ≤ dim V_{𝔖_{np}}(S^{η}↑^{𝔖_{np}}) = dim res_{𝔖_{np},𝔖_{np−1}}^{∗} V_{𝔖_{np−1}}(S^{η}) ≤ n − 1

the module S^{η}↑^{𝔖_{np}}↓_{E_{n}} is generically free. Hence S^{μ}↓_{E_{n}}, S^{ϕ(μ)}↓_{E_{n}} is a pair of modules of complementary stable Jordan type. Since S^{ϕ(μ)}↓_{E_{n}} has stable generic Jordan type either the same or complementary to that of S^{μ}↓_{E_{n}}, we get the second assertion. □

Lemma 5.9. Let n ≥ 2 be a positive integer and μ = (np − p, p). If p is an odd prime, then n − 2 ≤ n_{μ}(1) ≤ n + 1. If p = 2, then n_{μ}(1) = n − 2.

Proof. Let p be an odd prime. Note that S^{(np)}↓_{E_{n}} ≅ k_{E_{n}} has generic Jordan type (1) and S^{(np−1,1)}↓_{E_{n}} has stable generic Jordan type (p − 1), which is complementary to (1). The permutation module M^{(np−p,p)} decomposes into Q ⊕ S where S↓_{E_{n}} is a generically free kE_{n}-module and Q has three Specht factors S^{(np)}, S^{(np−1,1)} and S^{(np−p,p)} reading from the top. The stable generic Jordan type for M^{(np−p,p)}↓_{E_{n}} is given in Lemma 5.4 as (1^{n}). Suppose that A is the quotient Q/S^{(np−p,p)}. We want to figure out the possible stable generic Jordan types of A↓_{E_{n}}. Considering the short exact sequence 0 → S^{(np−1,1)}↓_{E_{n}} → A↓_{E_{n}} → k → 0 and using Proposition 2.2 (iii), we see that A↓_{E_{n}} is generically free or has stable generic Jordan type (p−1, 1). Now consider the short exact sequence 0 → S^{(np−p,p)}↓_{E_{n}} → Q↓_{E_{n}} → A↓_{E_{n}} → 0. In the case A↓_{E_{n}} is generically free, S^{(np−p,p)}↓_{E_{n}} has stable generic Jordan type the same as M^{(np−p,p)}↓_{E_{n}}, which is (1^{n}). Suppose that the stable generic Jordan type of A↓_{E_{n}} is (p − 1, 1). Using Proposition 2.2 (iii), one can figure out that the possible stable generic Jordan types of S^{(np−p,p)}↓_{E_{n}} are one of the listed below:

(a) (p − 1, 1^{n+1}),

(b) (p − 1, 2, 1^{n−1}),

(c) (1^{n}),

(d) (2, 1^{n−2}). In any case, n − 2 ≤ n_{μ}(1) ≤ n + 1.

Suppose that p = 2. Let α = (α_{1}, …, α_{n}) ∈ k^{n} be a generic point and u_{α} = 1 + ∑_{i=1}^{n} α_{i}((2i − 1, 2i) − 1). Consider the map

M^{μ}↓_{k⟨u_{α}⟩} = M_{1} ⊕ P_{1} ⟶^{f} S^{(2n−1,1)}↓_{k⟨u_{α}⟩} ⊕ S^{(2n)}↓_{k⟨u_{α}⟩} = M_{2} ⊕ P_{2}

given by f(t_{i,j}) = (t_{i} + t_{j}, 1) where t_{i,j} is the μ-tabloid with i, j in the second row and t_{i} is the (2n − 1, 1)-tabloid with i in the second row, also M_{1}, M_{2} are the non-free parts of M^{μ}↓_{k⟨u_{α}⟩}, S^{(2n−1,1)}↓_{k⟨u_{α}⟩} ⊕ S^{(2n)}↓_{k⟨u_{α}⟩} respectively. Note that the Jordan types of M_{1}, M_{2} are (1^{n}), (1^{2}) respectively, and P_{1}, P_{2} are projective modules or equivalently free modules. The kernel of the map f is precisely S^{μ}↓_{k⟨u_{α}⟩}. The module M_{1} is k-spanned by t_{2i−1,2i} for 1 ≤ i ≤ n; meanwhile, P_{1} is k-spanned by those t_{i,j}’s not listed before. The module M_{2} is k-spanned by both the tabloid in S^{(2n)} and ∑ t_{i} ∈ S^{(2n−1,1)} ⊆ M^{(2n−1,1)}. Consider the map f in the stable module category, the induced map f̃: M_{1} ↠ M_{2} splits and it has kernel with Jordan type (1^{n−2}). So ker f ≅ ker f̃⊕ P_{3} for some projective module P_{3}. This shows that the stable generic Jordan type of S^{μ}↓_{E_{n}} is (1^{n−2}), i.e., n_{μ}(1) = n − 2. □

5.3. Proof of Theorem 3.3. Suppose that μ is a partition satisfying Hypothesis 3.2. In order to prove Theorem 3.3, by Lemma 5.8, it suffices to show that V_{E_{n}}^{♯} (S^{Φ(μ)}) = V_{E_{n}}^{♯}(k). The idea of the proof is to show that n_{Φ(μ)}(r) > 0 for some 1 ≤ r ≤ p − 1.

Proof of Theorem 3.3. The result is obvious for Type (H3); indeed, n_{Φ(μ)}(1) = 1.

\underline{Type (H1)}: For any 1 ≤ m < p/2, let η(m) = (p^{2} − mp, mp) and λ(m) = (p^{2} − mp − 1, mp + 1). Note that Φ(λ(m)) = η(m). Taking n = p in Lemma 5.9, we have n_{η(1)}(1) ≥ p − 2. We claim that for all 1 ≤ m < p/2

n_{η(m)}(1) + n_{η(m)}(2) + … + n_{η(m)}(m) ≥ p − 2  (∗)

By induction on m, suppose that we have the inequality as in (∗). By Corollary 5.2, we have a decomposition

M^{η(m+1)} ≅ M^{η(m)} ⊕ S ⊕ F

where F ↓_{E_{p}} is a generically free kE_{p}-module and S has a filtration with Specht factors S^{λ(m)}, S^{η(m+1)} reading from the top. By Lemma 5.4, S↓_{E_{p}} has stable generic Jordan type 1^{\binom{p}{m+1}−\binom{p}{m}}. Note that S^{η(m)}↓_{E_{p}}, S^{λ(m)}↓_{E_{p}} is a pair of modules of complementary stable Jordan type, so S^{λ(m)}↓_{E_{p}} has stable generic Jordan type satisfying the inequality

n_{λ(m)}(p − m) + n_{λ(m)}(p − m + 1) + … + n_{λ(m)}(p − 1) ≥ p − 2

The short exact sequence 0 → S^{η(m+1)}↓_{E_{p}} → S↓_{E_{p}} → S^{λ(m)}↓_{E_{p}} → 0 leads to the short exact sequences 0 → J_{i} → J_{a} ⊕ J_{b} → J_{j} → 0 with a, b ∈ {0, 1, p}. We focus on the cases where p − m ≤ j ≤ p − 1. Note that we must have either J_{p} ⊕ J_{1} or J_{p} in the middle. Using Proposition 2.2 (iii), in the first case, i + j = 1 + p and hence 2 ≤ i ≤ m + 1; in the latter case, i + j = p and hence 1 ≤ i ≤ m. So we conclude that

n_{η(m+1)}(1) + n_{η(m+1)}(2) + … + n_{η(m+1)}(m + 1) ≥ p − 2

Since n_{η(m)}(i) is non-zero for some 1 ≤ i ≤ m, S^{η(m)}↓_{E_{p}} is not generically free. So the rank variety V_{E_{p}}^{♯}(S^{Φ(μ)}) is V_{E_{p}}^{♯}(k) in this case.

\underline{Type (H2)}: Note that n > 2. The proof is akin to the proof of Type (H1), except that we use Corollary 5.3 instead of Corollary 5.2 . So n_{Φ(μ)}(1)+n_{Φ(μ)}(2) ≥ n−2 > 0.

\underline{Type (H4)}: Consider the case where Φ(μ) = (2n−2, 2) ≠ (2, 2). Note that n > 2. Now we use Lemma 5.9 for p = 2 to deduce that n_{Φ(μ)}(1) = n − 2 > 0. Now take Φ(μ) = (2n − 4, 4) ≠ (4, 4) where n > 4. The permutation module M^{Φ(μ)} has a filtration with Specht factors S^{λ(i)} one for each 0 ≤ i ≤ 4 where λ(i) = (2n − i, i). Over p = 2, any short exact sequence of kE-modules 0 → A → B → C → 0 satisfies the inequality n_{B}(1) ≤ n_{A}(1) + n_{C}(1). So

\binom{n}{2} = n_{M^{Φ(μ)}↓_{En}}(1) ≤ ∑_{i=0}^{4} n_{λ(i)}(1)

Note that n_{λ(0)}(1) = n_{λ(1)}(1) = 1 and n_{λ(3)}(1) = n_{λ(2)}(1) = n − 2. Suppose that S^{Φ(μ)}↓_{E_{n}} is generically free, i.e., n_{Φ(μ)}(1) = 0, we deduce that n^{2} − 5n + 4 ≤ 0, i.e., 1 ≤ n ≤ 4, a contradiction. So S^{Φ(μ)}↓_{E_{n}} is not generically free, i.e., V_{E_{n}}^{♯}(S^{Φ(μ)}) = V_{E_{n}}^{♯}(k). □

Example 5.10. Let p be an odd prime and μ be the partition as in Example 5.6. Then V_{E_{p}}^{♯}(S^{μ}) = V_{E_{p}}^{♯}(k) and the complexity of S^{μ} is p.

Example 5.11. Let p = 3 and μ be a partition of 9. Suppose that μ ≠ (3^{3}). In the case μ̃ ≠ ∅, we use Proposition 2.1 (iv) and the hook formula to calculate the rank variety V_{E_{3}}^{♯}(S^{μ}); otherwise, we use Proposition 2.2 (v) and Theorem 3.3. In the case μ = (3^{3}), the software MAGMA was used by Jon Carlson to determine the rank variety V_{E_{3}}^{♯} (S^{(3^{3})}) which is precisely the zero set of the radical ideal

⟨x_{1}^{2}x_{2}^{2} + x_{2}^{2}x_{3}^{2} + x_{1}^{2}x_{3}^{2}⟩

Note that it fits into Theorem 3.1 (ii), where f̃ = 0 and n = 1.

The Variety V_{E_{3}}(S^{μ}) for the case p = 3 and |μ| = 9.

\begin{matrix}μ & V_{E_{3}}(S^{μ}) \\ (9), (1^{9}) & V_{3} \\ (81), (21^{7}) & V_{3} \\ (72), (2^{2}1^{5}) & V_{1} \\ (63), (2^{3}1^{3}) & V_{3} \\ (54), (2^{4}1) & V_{3} \\ (71^{2}), (31^{6}) & V_{3} \\ (621), (321^{4}) & V_{3} \\ (531), (32^{2}1^{2}) & \{0\} \end{matrix}  \begin{matrix}μ & V_{E_{3}}(S^{μ}) \\ (4^{2}1), (32^{3}) & V_{3} \\ (52^{2}), (3^{2}1^{3}) & V_{3} \\ (432), (3^{2}21) & V_{3} \\ (521^{2}), (421^{3}) & V_{1} \\ (431^{2}), (42^{2}1) & V_{1} \\ (3^{3}) & V_{2} \\ (61^{3}), (41^{5}) & V_{3} \\ (51^{4}) & V_{3} \end{matrix}

In Table 1, V_{1} = res_{E_{3},E_{1}}^{∗} V_{E_{1}}(k), V_{2} = V (x_{1}^{2}x_{2}^{2} + x_{2}^{2}x_{3}^{2} + x_{1}^{2}x_{3}^{2}) and V_{3} = V_{E_{3}}(k) where E_{1} = ⟨(1, 2, 3)⟩ ⊆ 𝔖_{9}. The subscript i of V_{i} gives the dimension, which is the complexity of S^{μ}.

Acknowledgement. I thank my supervisor David Benson for his guidance and suggestions in this paper.

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