ON CONJECTURES OF FOULKES, SIEMONS AND WAGNER AND STANLEY A thesis submitted to the School of Mathematics of the University of East Anglia in partial fulfilment of the requirements for the degree of Doctor of Philosophy By Thomas J. McKay July 2008 (cid:13)c This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with the author and that no quotation from the thesis, nor any information derived therefrom, may be published without the author’s prior, written consent. Abstract Let λ = (λ ,...,λ ) be a partition of n. An unordered λ-tabloid is a partition of the 1 r set{1,2,...,n}intor pairwisedisjointsetsofsizesλ ,...,λ . LetF denotethefield 1 r ofcomplexnumbersandGthesymmetricgroupof{1,2,...,n}. DefineHλ tobethe permutation module of FG whose basis is the set of unordered λ-tabloids. Foulkes conjectured in [13] that there exists an injective FG-homomorphism H(ba) → H(ab) when a ≤ b. Independently Siemons and Wagner [27] and Stanley [29] generalized this conjecture to ask if there exists an injective map Hλ → Hλ(cid:48). In this thesis we investigate these conjectures. ii Acknowledgements I would like to thank Johannes Siemons for all his help and support throughout the last seven years, for first introducing me to group theory and then for being my undergraduate/masters/PhD advisor. He has undertaken countless tasks, many of which he was under no obligation to, and his efforts are much appreciated. iii Contents Abstract ii Acknowledgements iii 1 Introduction and Overview 1 2 General results 5 2.1 Notation and definitions . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The space Hλ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Constructions of homomorphisms . . . . . . . . . . . . . . . . . . . . 10 2.4 The standard map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Lifting homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Generalized tabloids and Hom-spaces 24 3.1 Definitions and basic results . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 The (cid:15)-map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 The action of the twist group . . . . . . . . . . . . . . . . . . . . . . 30 3.4 The standard map ψ (x,y) . . . . . . . . . . . . . . . . . . . . . . . 31 (ba) 3.5 The standard map of (a,bc) . . . . . . . . . . . . . . . . . . . . . . . 33 4 Injective standard maps 36 4.1 Column removal and the modules Mλi . . . . . . . . . . . . . . . . . 36 4.2 Block removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 iv 5 Non-injective standard maps 56 5.1 The results of Sivek . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2 Good hooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.3 Restriction to S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 n−1 6 The standard map of partitions with at most four parts 75 6.1 Partitions with at most three parts . . . . . . . . . . . . . . . . . . . 75 6.2 Good partitions with four parts . . . . . . . . . . . . . . . . . . . . . 76 6.3 Bad partitions with four parts . . . . . . . . . . . . . . . . . . . . . . 78 v Chapter 1 Introduction and Overview Let λ = (λ ,...,λ ) be a partition of n. An unordered λ-tabloid is a partition of the 1 r set {1,...,n} into r pairwise disjoint sets of sizes λ ,...,λ . Let F denote the field 1 r of complex numbers and let G denote the symmetric group of the set {1,2,...,n}. Define Hλ to be the permutation module of FG whose basis is the set of unordered λ-tabloids. A long standing open problem is Foulkes’ Conjecture made in [13] that there exists an injective FG-homomorphism H(ba) (cid:44)→ H(ab) if a ≤ b. By explicitly computing the composition factors (see Example 9 on Page 140 of Macdonald’s Book [20]) one can easily see that this conjecture holds when a = 2 and b is arbitrary. Dent and Siemons [11] showed that the conjecture holds when a = 3 and b ≥ a by producing a linearly independent subset of Hom (Sλ,H(ab)) FG of size equal to the dimension of Hom (Sλ,H(ba)) for each irreducible FG-module FG Sλ that appears in H(ba). Briand [3] claimed to prove Foulkes’ Conjecture is true when a ≤ 4 and b ≥ a using diagonally symmetric functions although it is our understanding ([4] and [5]) that this proof contains a flaw and should be refined to a ≤ 3. In [6] and [7] Brion uses arguments from algebraic geometry to show that Foulkes’ Conjecture is true when b is large compared to a. In [2] Black and List defined a map ψ : H(ba) → H(ab) and conjectured that ba it was injective. Coker [9], Dent [10], Doran [12] and Pylyavskyy [23] then showed 1 that the map ψ is injective. If ψT denotes the adjoint of ψ , Coker and Dent b2 b2 b2 independently showed that all the eigenvalues of ψ ◦ ψT are non-zero. In [12] b2 b2 Doran observed that for each irreducible FG-module Sλ, the FG-homomorphism ψ induces an F-linear map ψˆ : Hom (Sλ,Hba) → Hom (Sλ,Hba) by θ (cid:55)→ ba ba FG FG T θ ◦ ψ . Using this he went on to show that ψ is injective. Pylyavskyy directly T ba b2 showed that the map ψ is injective using an inductive argument. Of particular b2 interest to us are the computational results of Jacob’s Thesis [15] and the paper by Mu¨ller and Neunho¨ffer [22]. In [15], Jacob proved that the Black and List map is injective when a = b = 2, a = b = 3 and a = b = 4. Mu¨ller and Neunho¨ffer [22] then showed that it is injective when a = b = 5. Independently, Siemons and Wagner [27] and Stanley [29] (SWS) extended the Black and List map in a natural way to an FG-homomorphism ψ : Hλ → Hλ(cid:48) λ and conjectured this to be injective iff λ dominates its conjugate. We call this the standard map. Whenλ = (r,1s)isahookthenthestandardmapisthemapbetween the layers M and M in the Boolean algebra of a set of size r +s. This is well r r+1 known (see Proposition 5.4.7 of Sagan’s book [24]) to be injective iff r > s and so the SWS conjecture holds in this case. In [23] Pylyavskyy shows that the standard map of (6,2,2,1,1) is not injective and refines the conjecture to say the standard map has maximal rank. In [28] Sivek shows that this new conjecture is also false and fails for large classes of partitions, the smallest being (4,3,3). Finally, in [30] Vessenes generalizes Foulkes’ Conjecture to conjecture that there exists an injective map H(ba) → H(dc) when a ≤ c and b ≥ d. The main result of [30] is then to prove that this conjecture holds when a = 2. In the setting of algebraic geometry Abdesselam and Chipalkatti [1] then proved that a given map ψ : H(b2) → H(dc) is injective when d,c ≥ 2. We now turn to our own efforts. This thesis has five chapters after the present introductoryone. TheimportantresultsofChapter2areProposition2.3.1andThe- orem 2.3.4 on pages 11 and 12. These results allow us to produce homomorphisms 2 between permutation modules with wanton abandon and view these as compositions of other homomorphisms. In Chapter 3 we look at spaces of homomorphisms be- tween tabloid spaces. We define the important (cid:15)-map and prove that it is injective. We also prove the following result: THEOREM 3.4.2 (page 33): Let V ∼= Sν be an irreducible submodule in the kernel of the standard map of (ba). Then ν has at least three parts. Chapter 4 is devoted to understanding when the standard map is injective. This leads to the following definition of a good column of a Young diagram on page 44. We say that the node λ is good if the hook h has an arm at least as long as its ij ij leg. We then say that a column is good if every node in it is good. We can now state the main theorem of the thesis THEOREM 4.1.15 (page 45): Let λ be a partition and µ the partition obtained by removing a good column from λ. Suppose the standard map of µ is injective. Then the standard map of λ is injective. When combined with the results of Jacob, Theorem 4.1.15 shows that Foulkes’ Conjecture holds when a ≤ 4. That is THEOREM 4.1.17 (page 45): Let a ≤ 4 and a ≤ b. Then the standard map ψ : H(ba) → H(ab) is injective. ba In Chapter 5 we look at when the standard map is not injective. Inspired by the definition of a good column, on page 62 we define A(i,j) to be the hook whose arm includes the ith highest removable node and whose leg includes the jth highest removable node. We say that a partition is good if all the A(i,j) have arm at least as long as their leg. Most of this chapter is then devoted to proving: THEOREM 5.2.4 (page 63): Suppose that the standard map of λ is injective. Then λ is good. In Chapter 6 we collect together all the results in the thesis to prove Theorem 6.1.3, which shows that the standard map controls the existence of injective maps Hλ → Hλ(cid:48) when λ has at most three parts. However when λ has four or more parts 3 the situation is more complicated as Theorem 6.3.14 shows THEOREM 6.1.3 (page 76): Let λ be a partition with at most three parts. Then the following are equivalent: (i) There exists an injective map Hλ → Hλ(cid:48). (ii) The partition λ is good. (iii) The standard map of λ is injective. THEOREM 6.3.14 (page 86): Let µ = (a,b+ 1,b2) with a ≥ 2b + 2. Then the standard map of µ is not injective but there exists an injective map Hµ → Hµ(cid:48). Theorem 5.2.4 tells us that a partition with an injective standard map is necce- sarily good, while Theorem 6.1.3 says that being good is sufficient for three part partitions to have an injective standard map. It is natural ask whether the property of being good is always sufficent. Sivek’s Lemma 5.1.7 states that adding rows to a partition with non-injective standard map yields a partition with non-injective standard map. Hence when combined with the result of Mu¨ller and Neunho¨ffer that ψ is non-injective we see that one can produce many partitions that are good (55) but have non-injective standard map. However something can be saved using our techniques. Let B(i,j) denote the unique hook whose arm lies in the highest row of length λ and whose leg contains the removable node in a row of length λ . In Sub- i j section 4.2.2 we strengthen the definition of good by saying that λ = (λm1,...,λmr) 1 r is extremely good if the arm of each B(i,j) is at least as long as its leg and for each i the standard map of (λ −λ )mi is injective and prove: i i+1 THEOREM 4.2.21 (page 55): Let λ be extremely good. Then the standard map of λ is injective. An interesting special case of Theorem 4.2.21 is the following: COROLLARY 4.2.22 (page 55): Let λ (cid:96) n. Suppose all the parts of λ are distinct. Then the standard map of λ is injective. Thus the answer to the SWS conjecture seems to lie somewhere between good and extremely good. 4 Chapter 2 General results 2.1 Notation and definitions A composition of a positive integer n is a finite sequence λ = (λ ,λ ,...,λ ) of 1 2 r positive integers such that (cid:80)r λ = n. A partition of n is a composition λ such i=1 i that λ ≥ λ for each i. We write λ (cid:96) n to identify λ as a partition of n. If a i i+1 partition has a parts of equal length b then we write (ba) instead of (b,b,...,b). For example we write (53,2,1) in place of (5,5,5,2,1). The Young diagram [λ] of a partition λ is the left aligned array of nodes such that the ith row has λ nodes. For example the Young diagram of (53,2,1) is i ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ [λ] = ◦ ◦ ◦ ◦ ◦ . ◦ ◦ ◦ The node in the ith row and jth column of [λ] is denoted by λ . The hook h i,j i,j consists of the node λ together with the λ −j nodes to the right of it (the arm) i,j i and the λ −i nodes below it (the leg). The arm length a of h is λ −j and the j i,j i,j i leg length l is λ −i. i,j j Example 2.1.1 Let λ = (53,2,1). Then a = 3 = l and h is illustrated below. 2,1 2,1 2,1 5