THE COMPLEXITY OF CERTAIN SPECHT MODULES FOR THE SYMMETRIC GROUP

DAVID J. HEMMER

Abstract. During the 2004-2005 academic year the VIGRE algebra research group at the University of Georgia computed the complexities of certain Specht modules S^{λ} for the symmetric group Σ_{d}, using the computer algebra program Magma. The complexity of an indecomposable module does not exceed the p-rank of the defect group of its block. The Georgia group conjectured that, generically, the complexity of a Specht module attains this maximal value; that it is smaller precisely when the Young diagram of λ is built out of p × p blocks. We prove one direction of this conjecture by showing these Specht modules do indeed have less than maximal complexity. It turns out that this class of partitions, which has not previously appeared in the literature, arises naturally as the solution to a question about the p-weight of partitions and branching.

1. Introduction

1.1. In 2004 the VIGRE algebra research group at the University of Georgia computed some examples of the complexity of Specht modules S^{λ} for the symmetric group Σ_{d}. Their data led them to focus on partitions λ of a curious form, specifically:

Definition 1.1. A partition λ ⊢ d is p × p if λ = (λ_{1}^{a_{1}}, λ_{2}^{a_{2}}, …, λ_{s}^{a_{s}}) where p | λ_{i} and p | a_{i} for all i.

Such λ can exist only if p^{2} | d. Equivalently, λ is p × p if both λ and its transpose λ′ are of the form pτ. Also equivalently, the Young diagram of λ is built from p × p blocks.

Now suppose S^{λ} is in a block B(λ) of weight w corresponding to a p-core λ̃ ⊢ d − pw. Then the defect group of B(λ) is isomorphic to a Sylow p-subgroup of Σ_{pw} and has p-rank w. In particular, the maximum complexity of any module in the block B(λ) is w. The VIGRE group made the following conjecture:

Conjecture 1.2 (UGA VIGRE ^{1}). Let S^{λ} be in a block B of weight w. Then the complexity of S^{λ} is w if and only if λ is not p × p .

Conjecture 1.2 implies that almost every Specht module has maximal complexity among modules in its block. Indeed it would imply that if p^{2} |̸ d, then all the Specht modules for Σ_{d} have this property. As far as we know the condition we call p×p has not appeared anywhere in the literature, and it seems quite mysterious. However we will prove that it arises quite naturally from considering the weights of Σ_{d} blocks and the branching theorems. Specifically we prove:

Date: November 2008.

2000 Mathematics Subject Classification. Primary 20C30.

Research of the author was supported in part by NSF grant DMS-0556260.

1This conjecture and some discussion can be found at http://www.math.uga.edu/∼ nakano/vigre/vigre.html.

Theorem 1.3. Suppose λ ⊢ d has p-weight w. Then λ is p × p if and only if w(λ_{A}) ≤ w − 2 for each removable node A of λ. In this case, w(λ_{A}) is always equal to w − 2.

Theorem 1.3 give one direction of Conjecture 1.2 as a fairly immediate corollary.

Corollary 1.4. Suppose λ ⊢ p^{2}d is p × p , and hence of weight w = pd. Then the complexity of S^{λ} is less than w.

Remark 1.5. When d < p^{2} Conjecture 1.2 can be deduced immediately from the dimensions of the corresponding Specht modules, as was noticed by the VIGRE group and proven by Lim [6, Thm. 4.1].

Remark 1.6. The conjecture was verified for d = p^{2} by Lim in [6, Thm. 3.1], where he showed S^{(p^{p})} has complexity p−1 even though (p^{p}) has weight p. Lim’s proof uses the fact that Res_{Σ_{p^{2}−1}}(S^{(p^{p})}) ≅

S^{(p^{p−1},p−1)} and (p^{p−1}, p − 1) has weight p − 2. Our proof is essentially a large generalization of this observation about the branching theorems.

2. Blocks and abaci

2.1. In this section we review the description of the blocks of kΣ_{d} and the representation of partitions of d on the abacus. For further details on this, and as good sources for the representation theory of the symmetric group, see the books [5] and [4], where definitions of basic terms not defined here, like removable node, addable node, residue of a node, etc.., can be found.

Let λ = (λ_{1}, λ_{2}, …, λ_{t}) ⊢ d with λ_{t} > 0. Choose an integer r ≥ t and define a sequence of beta-numbers by:

β_{i} = λ_{i} − i + r

for 1 ≤ i ≤ r. For r = s the β-numbers are just the first-column hook lengths in the Young diagram of λ. It is easily seen that λ can be recovered uniquely from a corresponding sequence of beta numbers.

Now take an abacus with p runners, labeled 0, 1, …, p − 1 from left to right. Label the positions on the abacus so that row s (counting down from the top) runner i is labeled i + (s − 1)p. To represent λ on the abacus with r beads, simply place a bead at position β_{i} for each 1 ≤ i ≤ r.

Sliding all the beads in the abacus display as far up as possible, one obtains an abacus display for the p-core λ̃. Moving a bead up a single spot corresponds to removing a rim p-hook from the Young diagram of λ. Thus, if w is the total number of such moves, then λ̃ ⊢ d − pw. The number w is called the p-weight of λ. The following proposition collects some well-known facts, all found in the book [5].

Proposition 2.1. Let λ ⊢ d have p-weight w, p-core λ̃ ⊢ d − pw, and exactly t parts. Let μ ⊢ d. Then:

(a.) The Specht modules S^{λ} and S^{μ} lie in the same p-block if and only if λ̃ = μ̃.

(b.) Suppose B(λ) is the block containing S^{λ}. Then a defect group of B(λ) is isomorphic to a

Sylow p-subgroup of the symmetric group Σ_{pw}. (c.) Removing a removable node from λ corresponds to sliding a bead in the abacus display of

λ from position l to (unoccupied) position l − 1. Adding an addable node corresponds to sliding a bead from position l to (unoccupied) position l + 1.

(d.) If λ is represented on an abacus with a multiple of p beads then removable nodes which

correspond to beads on runner i have p-residue i. (e.) Let {A_{1}, A_{2}, …, A_{m}} denote the removable nodes of λ and let λ_{A} ⊢ d − 1 denote λ with

removable node A removed. Then Res_{Σ_{d−1}}(S^{λ}) has a filtration by Specht modules where each S^{λ_{A_{i}}} occurs once.

2.2. Motivated by Proposition 2.1(d), we assume henceforth all our abacus displays have a multiple of p beads. Given such an abacus display for λ, let λ^{[i]} be the number of beads on runner i. Knowing these numbers is equivalent to knowing the p-core λ̃. For each runner we define a partition which captures how far each bead moves when sliding the beads of λ all the way up. Define λ_{j}^{i} as the number of empty spots above the jth bead from the top on runner i, i.e. the number of times this bead is moved up to obtain the p-core. Then λ(i) = (λ_{1}^{i}, λ_{2}^{i}, …) is a partition and the sequence of partitions (λ(0), λ(1), …, λ(p − 1)) is called the p-quotient of λ. It should be clear that λ is uniquely determined by its p-core and p-quotient, and that the parts of the p-quotient are partitions of integers which sum to w.

3. The main theorem

3.1. In this section we will prove Theorem 1.3. Our first observation is that the p × p condition is easily described in terms of the abacus display:

Proposition 3.1. Let λ ⊢ d have p-quotient (λ(0), λ(1), …, λ(p − 1)). Then λ is p × p if and only if λ̃ = ∅ and λ(0) = λ(1) = ⋯ = λ(p − 1).

Proof. This is a straightforward exercise. Suppose λ is p × p and write

λ = (pτ_{1}^{pa_{1}}, pτ_{2}^{pa_{2}}, …, pτ_{r}^{pa_{r}}).

Then λ has ∑_{i=1}^{r} pa_{i} parts and p-weight ∑_{i=1}^{r} pa_{i}τ_{i}. Representing λ on an abacus with ∑_{i=1}^{r} pa_{i} beads, one easily verifies that each λ(i) in the p-quotient is just (τ_{1}^{a_{1}}, τ_{2}^{a_{2}}, …, τ_{r}^{a_{r}}). Conversely given a partition with constant p-quotient λ(i) = τ and empty p core, one immediately observes that λ is p × p . □

Remark 3.2. The condition on λ in Proposition 3.1 is equivalent to each row of the abacus display being either empty or completely full. For example when p = 5 the partition λ = (20^{10}, 10^{5}, 5^{5}) has abacus display shown in Figure 3.1, where the corresponding τ is (4, 4, 2, 1).

[figure]
Figure 3.1. Abacus for p = 5, λ = (20^{10}, 15^{5}, 10^{15})

3.2. Suppose for the remainder of this section that λ ⊢ d has p-core λ̃ ⊢ c and weight w, so d = c + pw. Suppose further that for every removable node A, the partition λ_{A} which results from removing the node A satisfies w(λ_{A}) ≤ w − 2. We will show that λ has an abacus display as described in Remark 3.2, and thus that λ is p × p . Recall the notation λ^{[i]} for the number of beads on runner i, thus λ(i) ⊢ λ^{[i]}.

Lemma 3.3. Suppose λ has a removable node A of residue i > 0 Then λ^{[i−1]} ≥ λ^{[i]} + 1.

Proof. Removing a node corresponds to sliding a bead one spot to the left on the abacus. Suppose the bead corresponding to A is in row s of the abacus display, so there is no bead on runner i − 1 in row s. Further, suppose runner i − 1 has y beads in the first s − 1 rows and λ^{[i−1]} − y beads in rows > s. Assume runner i has x beads in the first s − 1 rows and λ^{[i]} − x − 1 beads in rows > s. Runners i − 1 and i in the abacus display for λ are illustrated in Figure 3.2.

[figure: i − 1; i; row s; y beads; x beads; bead of node A; λ^{[i−1]} − y beads; λ^{[i]} − x − 1 beads]

Figure 3.2. Removal of a node of residue i > 0

Then the weight of bead A changes from s − x − 1 to s − y − 1. In λ_{A}, the bottom λ^{[i−1]} − y beads on runner i − 1 are now weight one less and the bottom λ^{[i]} − x − 1 beads on runner i are weight one more. Thus the change in weight between λ and λ_{A} is:

(3.1)  w(λ_{A}) − w(λ) = s − y − 1 + λ^{[i]} − x − 1 − (s − x − 1 + λ^{[i−1]} − l)
= λ^{[i]} − λ^{[i−1]} − 1.

Our assumption on λ_{A} gives λ^{[i]} − λ^{[i−1]} − 1 ≤ −2, and thus λ^{[i−1]} ≥ λ^{[i]} + 1. □

Proposition 3.4. The number of beads on each runner in the abacus display of λ is nonincreasing from left to right, i.e. λ^{[i−1]} ≥ λ^{[i]}.

Proof. If λ^{[i−1]} < λ^{[i]} then the extra beads on runner i ensure λ has a removable node of residue i, which then contradicts Lemma 3.3. □

Next we consider the situation of a removable node of residue zero.

Lemma 3.5. Suppose λ has a removable node of residue zero. Then λ^{[0]} = λ^{[p−1]}.

Proof. Suppose λ has a removable node of residue zero, so the abacus display has a bead in row s+1 of runner zero, and no bead in row s of runner p−1. We let the reader draw the corresponding picture to Figure 3.2, and observe that

(3.2)  w(λ_{A}) − w(λ) = λ^{[0]} − λ^{[p−1]} − 2.

By Prop. 3.4 plus our assumption that w(λ_{A}) − w(λ) ≤ −2, we obtain the result. □

If any of the inequalities in Proposition 3.4 were strict, then we would have λ^{[0]} > λ^{[p−1]}, which would contradict Lemma 3.5. Thus we have proven that:

(3.3)  λ^{[0]} = λ^{[1]} ⋯ = λ^{[p−1]}.

Comparing equations (3.1) and (3.3), we discover that all removable nodes of λ have weight zero, since removable nodes of any other residue decrease the weight of the partition by only one. So the abacus display of λ has the same number of beads on each runner, and all the removable nodes have residue zero. This forces the abacus display to be as described in Remark 3.2, and completes the proof of Theorem 1.3. Notice that (3.2) and (3.3) implies w(λ_{A}) is always exactly w − 2, although our assumption was only that it is ≤ w − 2.

4. Complexity

4.1. Finally we prove Corollary 1.4, which gives one direction of Conjecture 1.2. A module has complexity c if the dimensions in a minimal projective resolution of that module are bounded by a polynomial of degree c − 1. The complexity of the trivial module is known to be the maximal rank of an elementary abelian subgroup, which for Σ_{pd} is just d. In general the complexity of a module for the symmetric group in a block of weight w is at most w. Good for this theory are [1] or [2].

So suppose λ ⊢ p^{2}d is p × p and so has weight w = pd. By Proposition 2.1(e) and Theorem 1.3, all summands of Res_{Σ_{p^{2}d−1}} S^{λ} lie in blocks of weight w − 2. By [3, 4.2.1d], Res_{Σ_{p^{2}d−1}} S^{λ} has complexity at most w − 2. An elementary argument using support varieties (see [6] for example) implies that S^{λ} cannot have complexity w. If it did, then for E a maximal elementary abelian subgroup of rank w, the support variety V_{E}(S^{λ}) would have dimension w, which would force the complexity of Res_{Σ_{p^{2}d−1}} S^{λ} to be w − 1.

Remark 4.1. For the case λ = p^{p}, dimension considerations immediately give the complexity of S^{(p^{p−1},p−1)}, which lets Lim deduce [6] that the complexity of S^{(p^{p})} is exactly p − 1. This part of the argument does not appear to generalize, i.e. in general it large powers of p can divide the dimension of S^{λ_{A}} when λ is p × p .

5. Problems

There are several obvious problems left unsolved.

Problem 5.1. Resolve the other direction of Conjecture 1.2.

Problem 5.2. Suppose λ is p × p of weight w. Is the complexity of S^{λ} equal to w − 1, or can it drop further?

Problem 5.3. One can generalize the definition of p × p . For example the first obvious generalization would be to require λ be p × p and each λ(i) be p × p . (And recursively for higher generalizations.) Can one say anything interesting about these situations? Perhaps the complexity drops by even more in this case?

References

[1] D. J. Benson. Representations and cohomology. II, volume 31 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1998. Cohomology of groups and modules.

[2] Leonard Evens. The cohomology of groups. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1991. Oxford Science Publications.

[3] David J. Hemmer and Daniel K. Nakano. Support varieties for modules over symmetric groups. J. Algebra, 254(2):422–440, 2002.

[4] Gordon James. The representation theory of the symmetric groups. Number 682 in Lecture Notes in Mathematics. Springer-Verlag, 1978.

[5] Gordon James and Adalbert Kerber. The representation theory of the symmetric group, volume 16 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Co., Reading, Mass., 1981.

[6] Kay Jin Lim. The varieties for some Specht modules. preprint, 2008.

Department of Mathematics, University at Buffalo, SUNY, 244 Mathematics Building, Buffalo, NY 14260, USA

   E-mail address: dhemmer@math.buffalo.edu