REPRESENTATIONS OF THE SYMMETRIC GROUPS AND COMBINATORICS OF THE FROBENIUS-YOUNG CORRESPONDENCE MATTHEW JOHN HALL

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1 REPRESENTATIONS OF THE SYMMETRIC GROUPS AND COMBINATORICS OF THE FROBENIUS-YOUNG CORRESPONDENCE By MATTHEW JOHN HALL A Thesis Submitted to The Honors College In Partial Fulfillment of the Bachelor s degree With Honors in Mathematics THE UNIVERSITY OF ARIZONA MAY 2008 Approved by: Christopher Ryan Vinroot Department of Mathematics

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3 Abstract After introducing the concepts of partitions, Young diagrams, representation theory, and characters of representations, the 2006 paper by A. M. Vershik entitled A New Approach to the Representation Theory of the Symmetric Group, III: Induced Representations and the Frobenius-Young Correspondence is discussed. In tracing through Vershik s line of reasoning, a flaw emerges in his attempt to prove one of the key lemmas. The attempted proof is an induction argument which, if valid, would lead to a purely combinatorial proof of the Frobenius-Young correspondence. Vershik asserts that two statements are equivalent, while in fact the implication is true in only one direction. Counterexamples are then given to Vershik s combinatorial statement and are then translated into counterexamples in representation theory. Finally, possible directions for further study are noted.

4 CONTENTS 4 Contents 1 Introduction 5 2 Partitions Conjugate Partitions Diagrams and Tableaux Orderings Dominance Ordering on Conjugate Partitions Representation Theory Group Algebras CG-modules Irreducible Representations Decomposition of Representations Restricted and Induced Representations Characters Inner Products Frobenius Reciprocity The Number of Irreducible Representations Double Cosets Mackey s Theorem Representations of the Symmetric Groups The Irreducible Representations of S n Branching Rule Young Subgroups Frobenius-Young Correspondence Mackey s Theorem Applied to S n The Snapper Conjecture Combinatorial Statement Combinatorial Counterexamples Representation Theory Counterexamples Frobenius-Young Correspondence, Revisited Further Questions 28

5 1 INTRODUCTION 5 Prelude The theory of representations of finite groups focuses on homomorphisms from a finite group to the group of invertible linear transformations of a vector space. After spending a semester actively studying these areas with the help of the excellent text Representations and Characters of Groups by Gordon James and Martin Liebeck, I was encouraged by my thesis adviser, Professor Ryan Vinroot, to explore a recent journal article by the mathematician A. M. Vershik entitled A New Approach to the Representation Theory of the Symmetric Groups, III: Induced Representations and the Frobenius-Young Correspondence, since this article served to illustrate and apply many of the concepts I had been learning. While making my way through this dense work, I attempted to prove for myself some of the theorems Vershik gave, so as to better understand his reasoning. However, after a rather lengthy struggle to prove a central lemma of the paper, I realized that there were in fact counterexamples, and hence the proof of the lemma was not valid. After discussing the problem with Ryan, who agreed that my counterexamples held, I began to investigate the ramifications of this error, whether or not the paper s main argument could still stand, and whether the error could be corrected. This thesis is the culmination of these efforts; through it, I hope to give to the reader familiar with the basic concepts of group theory a mathematical background sufficient to understand Vershik s paper, and then to review his argument and the counterexamples. 1 Introduction This paper begins with an overview of some broad notions from combinatorics, in particular the concepts of partitions of positive integers, Young diagrams, Young tableaux, and orderings on the set of Young diagrams. These tools are foundational to understand Vershik s argument in [10], because he attempts a purely combinatorial proof of the Frobenius-Young correspondence (see Section 4.4). Next, some basic concepts of the representation theory of finite groups are discussed, including Maschke s Theorem and its consequences, induced representations, the Frobenius Reciprocity Theorem, the number of non-isomorphic irreducible representations, and Mackey s Theorem. These tools are necessary to understand what the Frobenius-Young correspondence states, as well as to translate combinatorial counterexamples to one of Vershik s statements into their equivalent representation theoretic counterexamples. Finally, the representation theory of the symmetric group S n is explored, using both methods from both combinatorics

6 2 PARTITIONS 6 and representation theory of general finite groups. After a formal statement of the Frobenius-Young correspondence, facts such as the branching rules and the notion of Young subgroups are used to trace through Vershik s argument. An in-depth analysis of Vershik s third lemma (see section 4.6) reveals that where Vershik claims an equivalence between statements, there is in fact only an implication in one direction. This error is explored, an infinite family of combinatorial counterexamples is given, and the equivalent counterexamples in representation theory are also given. Ultimately, though Vershik s attempted purely combinatorial proof collapses, it is possible that there may be another method by which to successfully complete such a proof, and this question is proposed for further investigation. 2 Partitions Partitions are an important tool in group theory, particularly in the theory of the symmetric group. A partition λ of the integer n, written λ n, is a sequence of positive integers λ = (λ 1, λ 2,..., λ k ) such that λ i λ i+1 for all integers 1 i (k 1) and k i=1 λ i = n. Each element λ i of the partition is called a part of the partition. 2.1 Conjugate Partitions For each partition λ of n, there is a conjugate partition of n, denoted λ = (λ 1, λ 2,..., λ l ), where λ i = max{j λ j i}. The fact that λ is actually a partition of n may not be immediately apparent but will become clearer with the introduction of Young diagrams. 2.2 Diagrams and Tableaux One important way of visualizing partitions is through the use of Young diagrams. For each partition λ = (λ 1, λ 2,..., λ k ) the corresponding Young diagram, [λ], is a left-justified diagram in which the ith row from the top contains λ i boxes, or cells. For example, the Young diagram of the partition λ = (6, 4, 1, 1) of n = 12

7 2 PARTITIONS 7 is: [λ] = The conjugate partition, as defined above, yields a conjugate Young diagram. In relation to the original Young diagram [λ], the conjugate Young diagram [λ ] turns out to be the diagram which interchanges the rows and columns of [λ]; that is, the diagram obtained by reflecting [λ] about the main diagonal. This helps to justify the fact that the conjugate partition is in fact another partition of n, since by considering the Young diagrams, one may realize that reflection about the main diagonal will result in a new diagram with the same number of boxes, namely n. In particular, if λ = (6, 4, 1, 1) as above, the conjugate partition is λ = (4, 2, 2, 2, 1, 1) and the Young diagram associated with this partition is: Proposition 2.1. (λ ) = λ [λ ] = Proof. Passing to the conjugate is equivalent to making the rows of the original diagram into the columns of the conjugate diagram and vice versa. Thus, taking the conjugate again will switch the rows and columns back, once again giving the original diagram. Sometimes, such as when working with the certain aspects of the symmetric group S n, it may be desirable to place the integers 1 through n, once each, in the boxes of the Young diagram; the result of this is a Young tableau. Using the aforementioned partition λ = (6, 4, 1, 1) of n = 12, one Young tableau of this partition is given as follows: One other important term is the shape of a Young tableau, which refers to the partition that corresponds to the underlying Young diagram. The Young tableau

8 2 PARTITIONS 8 above has shape λ = (6, 4, 1, 1), for example. 2.3 Orderings There are two important orderings on Young diagrams, and thus also on partitions of n. The dominance ordering is a partial ordering on partitions of n. Given any two partitions µ = (µ 1, µ 2,..., µ l ) and λ = (λ 1, λ 2,..., λ k ) of n, µ dominates λ, denoted µ λ or λ µ, if and only if m i=1 µ i m λ j for all m between 1 and max{k, l}, inclusive, defining µ i = 0 if i > l and λ j = 0 if j > k. In terms of Young diagrams, [µ] [λ] if and only if for all m between 1 and max{k, l}, there are more boxes in the first m rows of [µ] than there are in the first m rows of [λ]. For instance, given the partitions µ = (6, 3, 2, 1) and λ = (5, 3, 3, 1) of n = 12, one can see that µ λ, that is: For n 5, the dominance ordering is more than a partial ordering; it is in fact a total ordering. Consider, for instance, the first two values of n. When n = 1, the only Young diagram is the unique diagram with one block, that is: [λ 1 ] = Moving up to n = 2, there are two distinct Young diagrams: the one obtained by adding a second block to the one row of [λ 1 ] and the one obtained by adding a second row of one block to [λ 1 ]. It is clear that this is total ordering, since By constructing the tree of Young diagrams which partition successively greater n, it can be checked that for n 5, any two diagrams partitioning n can be compared by the dominance ordering; hence, for these n, the dominance ordering is a total ordering. However, when n = 6, the partitions and corresponding diagrams of λ = (3, 1, 1, 1) and µ = (2, 2, 2) cannot be compared with the dominance ordering, and so the ordering becomes only a partial ordering. The other important ordering is used to relate two Young diagrams which correspond to partitions of integers n and n + 1. If γ n and λ (n + 1), then

9 2 PARTITIONS 9 one writes that γ λ or λ γ if and only if the diagram [λ] can be obtained by adding one block to [γ] in such a way that the result is still a valid diagram, or equivalently, [γ] can be obtained by removing a block from [λ] in such a way that the result is still a valid diagram. Using the partition λ = (6, 3, 2, 1) of 12, and defining a partition γ = (5, 3, 2, 1) of 11, one can see by looking at the Young diagrams that γ λ: 2.4 Dominance Ordering on Conjugate Partitions In Vershik s attempt to develop a purely combinatorial proof of the Frobenius- Young correspondence (see Section 4.4), he seeks to build a connection between two collections of Young diagrams. His goal in doing so is to eventually show the uniqueness of a diagram that he claims is common to both collections. The proof of Vershik s second lemma is a corollary of another lemma, which follows. Lemma 2.2. Let λ and µ be two partitions of n. λ µ. Then µ λ if and only if Proof. (Following the proof of (1.11) in [4].) First, note that proving one implication also proves the other, because if (µ λ) (λ µ ), then Proposition 2.1 implies that (λ µ ) (µ λ ) (µ λ). Define partitions λ = (λ 1, λ 2,..., λ a ) and µ = (µ 1, µ 2,..., µ b ) of the integer n. Suppose λ µ and µ λ (this second condition is necessary; without it, one could simply switch the roles of µ and λ). Then there exists an j N such that p λ i i=1 p i=1 µ i for all 1 p j 1, and j λ i < i=1 j µ i. (1) Taken together, these two equations imply that µ j = m is greater than λ j = l. Also, since the sum of all the parts of both µ and λ must equal n, equation (1) i=1

10 2 PARTITIONS 10 implies that a i=(j+1) λ i > b i=(j+1) µ i. (2) Now consider the Young diagrams [λ] and [µ] corresponding to λ and µ, respectively. The sum a i=(j+1) λ i gives the number of boxes of [λ] which are to the right of column j, and similarly the sum b i=(j+1) µ i gives the number of boxes of [µ] which are to the right of column j. Hence, a λ i = l (λ k j) and b µ i = (µ k j). i=(j+1) k=1 i=(j+1) k=1 Thus, applying equation (2) and observing that m > l and µ k j whenever 1 k m, one may see that l (λ k j) > k=1 (µ k j) k=1 l (µ k j). (3) k=1 Adding lj to both sides of the inequality of the leftmost and rightmost sum in equation (3) yields l l λ k > k=1 and thus, µ λ. Therefore, (λ µ ) (µ λ), and the contrapositive of this statement shows that (µ λ) (λ µ ), which by the earlier argument also demonstrates that (µ λ) (λ µ ). Corollary 2.3. (Lemma 2 in [10]) Let [λ] be an arbitrary diagram which partitions n, and let [λ ] be its conjugate diagram. Define two collections of diagrams: A = {[α] α n, α λ} and B = {[β ] β n, β λ }. Then A B = {[λ]}. Proof. By the above lemma, B is equivalent to the set of all diagrams [β ] for which λ β (since λ = λ by Proposition 2.1). So for a diagram [φ] to be in the intersection of A and B, it must be true that both φ λ (to be in A) and λ φ (to be in B). Since is a partial ordering, this can only occur if φ = λ. Thus, [λ] is the only diagram common to both sets. k=1 µ k

11 3 REPRESENTATION THEORY 11 3 Representation Theory Let G be a finite group. A representation of G over F is a homomorphism φ from a group G to GL(n, F) for some field F. In this paper, the field F will be taken to be the field of complex numbers, C, so a representation will be a map φ : G GL(n, C) such that φ(gh) = φ(g)φ(h) for all g, h G. The value of n is called the dimension of the representation φ. All groups G have a representation φ : G GL(1, C) where φ(g) = 1 for all g G. This representation is called the trivial representation of G and is denoted 1. For the group G = S n, one other important representation of dimension 1 is the sign representation, denoted sgn, for which { 1 if g is an even permutation, sgn(g) = 1 if g is an odd permutation. Two representations ρ : G GL(m, C) and σ : G GL(n, C) are said to be equivalent if and only if m = n and there exists a matrix T GL(n, C) such that σ(g) = T 1 (ρ(g))t for all g G. In this case, one writes ρ = σ. 3.1 Group Algebras Given a finite group G and the field C, one useful vector space is the group algebra of G over C, denoted CG. This vector space is the set of all formal C-linear combinations of elements in G, that is, { } CG = α g g α g C. g G The group algebra CG has addition defined by ( ) ( ) α g g + β g g = g + β g )g g G(α g G g G g G and multiplication defined by ( ) ( ) α g g β h h = α g β h (gh). h G g,h G

12 3 REPRESENTATION THEORY CG-modules Given a group G and a finite-dimensional vector space V over the field C, V is a CG-module if there exists a multiplication gv which satisfies the following five properties for all g, h G, u, v V, c C: (1) gv V (2) (gh)v = g(hv) (3) 1v = v (4) g(cv) = c(gv) (5) g(u + v) = gu + gv Theorem 3.1. CG-modules correspond exactly to representations of G over C. Proof. Given a group G, a representation ρ : G GL(n, C), and V = C n, a CG-module can be obtained by defining multiplication gv as gv = (ρ(g))v for all g G, v V. One can check that the five axioms hold for this definition, and thus the representation gives rise to a CG-module. To prove the correspondence in the other direction, consider a CG-module V with basis B = {v 1, v 2,..., v n }. For each g G, define by [g] B the matrix corresponding to the map f : V V such that f(v) = gv, relative to the basis B. Then since (gh)v = g(hv) for all g, h G and all v B, [gh] B = [g] B [h] B. Furthermore, since this implies that [1] B = [g] B [g 1 ] B, each matrix [g] B is invertible. Thus, the map φ for which φ(g) = [g] B is a homomorphism from G to an invertible matrix, that is, φ : G GL(n, C) (where n = dim V ), and so φ is a representation of G over C. For a given CG-module V, a CG-submodule of V is a subspace W of V which is also a CG-module under the same multiplication. Note here that for any CGmodule V, both the zero subspace {0} and the whole space V are CG-submodules. Now define V to be the vector space CG, with the multiplication of CG on V defined in the natural way. Then, since the five axioms are satisfied, V is a CG-module, called the regular CG-module. 3.3 Irreducible Representations A CG-module V is called irreducible if V {0} and the only CG-submodules of V are {0} and V ; otherwise, V is said to be reducible. Likewise, a representation

13 3 REPRESENTATION THEORY 13 ρ : G GL(n, C) is irreducible if and only if the corresponding CG-module C n given by gv = (ρ(g))v for all g G, v C n is irreducible. Any representation which is not irreducible is also said to be reducible. 3.4 Decomposition of Representations One of the most important theorems in representation theory (see [2], Chapter 8) is Maschke s Theorem, which implies that any representation can be decomposed into a direct sum of irreducible representations. Theorem 3.2. (Maschke s Theorem [over C]) Let G be a finite group and V be the corresponding CG-module. Then if U is any CG-submodule of V, there exists a CG-submodule W of V such that V = U W. A CG-module V is said to be completely reducible if V = k i=1 U i where each U i is an irreducible CG-submodule of V. Using Maschke s Theorem and induction on dim(v ), one can prove the following important fact: if G is a finite group, then every nonzero CG-module is completely reducible. 3.5 Restricted and Induced Representations If ρ is a representation of a group G and H is a subgroup of G, then the restricted representation, Res H ρ, is a representation of H obtained by defining (Res H ρ)(h) = ρ(h) for all h H. In the same sense that restricted representations yield information about representations of a smaller group from information about representations of a larger group, one may also consider induced representations, which use information about a subgroup to give information about representations of the whole group. However, before defining an induced representation, new notation must be established. Given a finite group G, consider a subset X of CG and define a subspace X(CG) of CG by X(CG) = span C {gx : g G, x X}.

14 3 REPRESENTATION THEORY 14 Then X(CG) is a CG-submodule of CG. Now let H be a subgroup G. If U is a CHsubmodule of CH, then the induced module from H to G is the CG-module U(CG). This CG-module corresponds to a representation called the induced representation, which is denoted by Ind G Hρ where ρ is the representation of H corresponding to U. 3.6 Characters For each representation φ of a group G, define the character χ of the representation to be the function χ : G C such that χ(g) = tr(φ(g)) for all g G. A central theorem of representation theory, the proof of which may be found in any standard text on the subject (for example, Proposition 13.5(1) and Theorem of [2]), is the following surprising and useful fact: Theorem 3.3. Suppose ρ and σ are two representations of a group G over C with characters χ and ψ, respectively. Then ρ is equivalent to σ if and only if χ(g) = ψ(g) for all g G. Given a finite group G which has a representation ρ with character χ and a subgroup H of G which has a representation σ with character ψ, the character of the representation Res H ρ obtained by evaluating χ on only the elements of H is called the restriction of χ to H and is written Res H χ. Similarly, the character of Ind G Hσ is called the induction of ψ to G, written Ind G Hψ. Let 1 denote both the trivial representation of a group as well as the character of the trivial representation, which are essentially the same, since the trivial representation has dimension one. As proved by Proposition of [2], this character has values given by the equation (Ind G Hψ)(g) = 1 H y G ψ(y 1 gy) for all g G, where the value of ψ is given as follows: { ψ(g) if g H, ψ(g) = 0 if g H. Using this formula, one can prove the following fact: Theorem 3.4. Let G be a group with subgroup H, and let g G. Then Ind G H1 = Ind G ghg 11.

15 3 REPRESENTATION THEORY 15 Proof. Consider the value of the induced characters on an arbitrary element x G. By the above formula, (Ind G ghg 11)(x) = 1 1 ghg 1 ghg 1(y 1 xy). y G Since conjugation of a subgroup does not change its order, and since y 1 xy ghg 1 implies y 1 xy H, (Ind G ghg 11)(x) = 1 1 H (y 1 xy). H y G As y runs over G, yg also runs over G, and so (Ind G ghg 11)(x) = 1 1 H ((yg) 1 x(yg)) = 1 1 H (g 1 y 1 xyg). H H y G y G But y 1 xy ghg 1 g 1 y 1 xyg H, which implies that 1 H ((yg) 1 x(yg)) will be nonzero exactly when 1 ghg 1(y 1 xy) is nonzero. Hence, (Ind G ghg 11)(x) = (IndG H1)(x), and so the characters of the induced representations agree. Therefore, by Theorem 3.3, the induced representations are equivalent. 3.7 Inner Products Another important concept in representation theory is that of an inner product. In general, an inner product associates to any two vectors φ, ψ from a given vector space V a number φ, ψ from a given field F such that the following three properties hold: (1) φ, λ = λ, φ for all φ, λ V (2) c 1 φ 1 + c 2 φ 2, λ = c 1 φ 1, λ + c 2 φ 2, λ for all c 1, c 2 F and φ 1, φ 2, λ V (3) φ, φ > 0 for all φ 0 Looking in particular at the vector space of functions from a group G to the field C, an inner product of two characters χ, ψ of G may be defined as follows: χ, ψ = 1 G χ(g)ψ(g 1 ). g G

16 3 REPRESENTATION THEORY 16 It is straightforward to check that this definition satisfies the three properties that define an inner product. Even more can be said about taking the inner product of two irreducible characters, that is, characters which correspond to irreducible representations. As shown in the proof of Theorem in [2], if χ, ψ are irreducible characters of a finite group G, then { 1 if χ = ψ, χ, ψ = 0 if χ ψ. This important equation is called the Schur orthogonality relation. Finally, since any representation may be decomposed into a direct sum of irreducible representations by Theorem 3.2, any character of G can be written as a sum irreducible characters. That is, if {χ 1, χ 2,..., χ k } is the set of all irreducible characters of G (see Theorem: 3.6), then an arbitrary character ψ of G can be written as ψ = k i=1 m χ i χ i for some nonnegative integers m χi for which m χi = ψ, χ i. By extension, this also shows that to the inner product of any two characters of G must be an integer (see [2], Theorem 14.24). This integer gives information about the multiplicities of the irreducible representations common to the decompositions of the two representations corresponding to the characters in the inner product. 3.8 Frobenius Reciprocity One very elegant theorem of representation theory is the Frobenius Reciprocity Theorem, which describes an important relationship between restricted and induced representations. A proof of this theorem may be found in [6, Theorem ]. If H is a subgroup of G, χ is a character of G, and ψ is a character of H, then the Frobenius Reciprocity Theorem states: Theorem 3.5. (Frobenius Reciprocity Theorem) Ind G Hψ, χ G = ψ, Res H χ H. 3.9 The Number of Irreducible Representations For a given group G with an element x, recall that the conjugacy class of x in G is the set x G = {g 1 xg g G}.

17 3 REPRESENTATION THEORY 17 For any finite group G, it is possible to determine the exact number of (nonisomorphic) irreducible representations (or, equivalently, characters) of G via the following fact, the proof of which may be found in Chapter 15 of [2]: Theorem 3.6. The number of distinct irreducible characters of G is equal to the number of distinct conjugacy classes of G Double Cosets Before stating the last theorem of representation theory needed for the remainder of this paper, it is necessary to understand double cosets. Given an element s in a group G, which has subgroups H, K, the (H, K)-double coset of s in G is the set HsK = {hsk h H, k K}. Theorem 3.7. Define a relation by x y if and only if x and y are in the same (H, K)-double coset, that is, y = hxk for some h H, k K. Then is an equivalence relation. Proof. (Reflexive:) Let x G. Then x = 1x1 and, since H and K are subgroups, 1 H and 1 K; thus x x. (Symmetric:) Let x, y G such that x y. Then y = hxk for some h H and k K, so x = h 1 yk 1. But h 1 H and k 1 K, and hence y x. (Transitive:) Let x, y, z G such that x y and y z. Then y = h 1 xk 1 and z = h 2 yk 2 for some h 1, h 2 H and k 1, k 2 K. Then z = h 2 (h 1 xk 1 )k 2 = (h 2 h 1 )x(k 1 k 2 ), and since h 2 h 1 H and k 1 k 2 K, x z. Hence, just like left or right cosets, the (H, K)-double cosets of G form a partition of G. Furthermore, the left and right cosets are a sub-partition of the (H, K)-double cosets. To see this, recall that y = xk for some k K if and only if x and y are in the same left coset of K in G, or y xk. But then it is also true that y = 1xk, and since 1 H, this implies that y HxK, and thus x and y are in the same (H, K)-double coset. A similar argument shows the same result for right cosets Mackey s Theorem One question one might ask is what happens when a representation is induced from one subgroup up to a larger group and then the resulting representation is restricted back down to another subgroup. The answer to this question is given by an important theorem known as Mackey s theorem. Let G be a finite group,

18 4 REPRESENTATIONS OF THE SYMMETRIC GROUPS 18 H and K be subgroups of G, and ρ be a representation of H. Given s in the set of (H, K)-double coset representatives, define H s = shs 1 K and define the representation ρ s of H s by ρ s (x) = ρ(sxs 1 ). Mackey s theorem, a proof of which may be found in Section 7.3 of [7], then states the following: Theorem 3.8. (Mackey s Theorem) Res K Ind G Hρ = s K\G/H Ind K H s ρ s. 4 Representations of the Symmetric Groups With an understanding of the representation theory presented above, it is now possible to begin looking at a recent paper by A. M. Vershik. His 2006 article entitled A New Approach to the Representation Theory of the Symmetric Groups, III: Induced Representations and the Frobenius-Young Correspondence attempts to use a purely combinatorial argument to present a new proof of a previouslyknown theorem. The aforementioned article draws on earlier articles written by Vershik and Okounkov (see [5], [9]). In the introduction to [9], Vershik and Okounkov state that they seek a more direct and natural method of realizing the main combinatorial ideas of the representation theory of S n. Specifically, the paper developed a new way of looking at the representations of S n which considers the whole chain S 1 S 2 S n and builds inductively to S n+1. Perhaps the most important difference between Vershik and Okounkov s method in contrast to the older method is that the new perspective gives the branching rules (see Section 4.2, below) immediately as a natural result rather than as a final corollary of a quite technical argument. Relying on these same ideas (the chain of embedded symmetric groups and the branching rules), Vershik strives in [10] to develop a purely combinatorial proof of the famous Frobenius-Young correspondence (see Section 4.4). This correspondence is known to be true from earlier proofs, such as the one given in Lecture 4 of [1], but Vershik desires to reach the same conclusion via a different (inductive) method. However, as will soon be demonstrated, Vershik s argument is flawed; in his sequence of lemmas, Vershik asserts that two statements are equivalent when in fact the implication only is true in one direction. This mistake invalidates Vershik s attempted proof, as shown through various counterexamples.

19 4 REPRESENTATIONS OF THE SYMMETRIC GROUPS The Irreducible Representations of S n It is known (see, for instance, Chapter 12 of [2]) that for the group G = S n, conjugacy classes correspond to cycle-type, that is to say: Theorem 4.1. For x S n, the conjugacy class x Sn consists of all permutations in S n which have the same cycle-type as x. Cycle-types in S n correspond to partitions of the integer n. Hence, applying Theorem 3.6, the number of irreducible characters of S n is equal to the number of partitions of n. Thus, for any partition λ n, one can define an irreducible representation of S n, denoted by π λ, which corresponds to the given partition. There are several methods to construct these π λ, one of which was the method developed by Vershik and Okounkov in [5] and [9], and it is this point of view which will be assumed for the remainder of this paper. By way of example, consider the partition λ = (3, 2, 1) of n = 6, which corresponds to the cycle-type (3,2,1), which in turn corresponds to the conjugacy class of S 6 of all elements with cycle-type (3,2,1), which in turn corresponds to some irreducible representation π λ of S Branching Rule A central idea of Vershik s two earlier papers is the development of a known theorem (for example, see Theorem of [6]) which gives information about the restriction or induction of an irreducible representation ρ of S n to S n 1 or S n+1, respectively. This theorem is key to Vershik s overall proof strategy because he argues using induction on n, and so the branching rule gives him a way to move from S n up to S n+1. Theorem 4.2. (Branching Rule) If λ n, then Res Sn 1 π λ = π γ, and 4.3 Young Subgroups Ind S n+1 S n γ λ π λ = π µ. For each symmetric group S n, a subgroup is called a Young subgroup if it is isomorphic to 1 2 k for some partition λ = (λ 1, λ 2,..., λ k ) of the λ µ

20 4 REPRESENTATIONS OF THE SYMMETRIC GROUPS 20 integer n. One specific Young subgroup, denoted by, is the Young subgroup of S n for which 1 acts on the set {1, 2,..., λ 1 } i acts on the set {( i 1 ) λ j + 1,. ( i 1 ) λ j + 2,..., ( i 1 ) } λ j + λ i k acts on the set {( k 1. ) λ j + 1, ( k 1 ) } λ j + 2,..., n. ( ) n Now, for any m n, there are ways to embed the group S m within S n m such that both symmetric groups act on individual elements (as opposed to pairs of elements or subgroups, for instance), but for the remainder of this paper, the assumption will be that an element of S m will act on the numbers 1 through m and leave the remaining (n m) elements fixed. Based on facts known from group theory about the conjugation of subgroups, given an element σ S n, the Young subgroup σ σ 1 is isomorphic to the Young subgroup. For this conjugate subgroup, the 1 term in the direct product expansion acts on the elements {σ(1), σ(2),..., σ(λ 1 )}, and similarly for any i, the i term in the direct product expansion acts on the elements { ([ i 1 ] ) ([ i 1 ] ) ([ i 1 ] )} σ λ j + 1, σ λ j + 2,..., σ λ j + λ i. Also note that given any partition λ = (λ 1, λ 2,..., λ k ) of n, each Young subgroup isomorphic to 1 2 k corresponds to a Young tableau of shape λ by filling in the row of [λ] corresponding to the part λ i with the subset of {1, 2,..., n} on which the i factor acts in the Young subgroup. The tableau corresponding to a subgroup will be denoted with brackets; for instance, the tableau for is [ ]. In any of these Young tableaux, the convention will be to write the numbers in a given row in strictly increasing order from left to right and, if multiple rows have equal length, to order them so that the rightmost boxes of these rows are strictly increasing from top to bottom. Finally, note that if any two Young tableaux have

21 4 REPRESENTATIONS OF THE SYMMETRIC GROUPS 21 the same shape, then they must arise from isomorphic Young subgroups, both isomorphic to, where λ is the shape of the Young tableau. To help illustrate these concepts, consider the partition λ = (6, 3, 2, 1) of n = 12. A corresponding Young subgroup will then be isomorphic to S 6 S 3 S 2 S 1. The Young subgroup can then be translated into a Young tableau; given some transposition σ = (3, 12) S 12, for example, [ ] = and [σ σ 1 ] = As is the case in this particular example, the Young tableau corresponding to any will have the integers 1 through n in successive order moving from left to right and top to bottom. 4.4 Frobenius-Young Correspondence Using the definitions of the previous subsection, consider the sets of irreducible components of the decompositions of Ind Sn 1 and Ind Sn sgn with multiplicity. The Frobenius-Young correspondence, given as Corollary 4.39 in [1], asserts that these two sets have exactly one representation in common, which is π λ, and that this representation occurs in each set with multiplicity one. Theorem 4.3. (Frobenius-Young correspondence) If the decomposition of Ind Sn 1 into irreducible representations is p i=1 ρ i and the decomposition of Ind Sn sgn into irreducible representations is q γ j, then {ρ i 1 i p} {γ j 1 j q} = {π λ }. 4.5 Mackey s Theorem Applied to S n In his attempt to prove the Frobenius-Young correspondence using the branching rule, Vershik uses a sequence of three lemmas from which the desired correspondence is supposed to follow. The first of these is an application of Mackey s Theorem (Theorem 3.8) to the symmetric group. Theorem 4.4. (Lemma 1 in [10]) Res Sn 1 Ind Sn 1 = γ:γ λ c(λ, γ)ind S n 1 S γ 1

22 4 REPRESENTATIONS OF THE SYMMETRIC GROUPS 22 where c(λ, γ) is the number of rows of the diagram [λ] which have the same length as the row of [λ] which is being modified to yield the diagram [γ]. Proof. Mackey s Theorem yields the following (taking G = S n, H =, K = S n 1, and ρ equal to the trivial representation 1): Res Sn 1 Ind Sn 1 = σ S n 1 \S n/ Ind Sn 1 σ σ 1 S n 1 1. Note that ρ s = 1 since the value of the trivial character is 1 for all g G, and hence for all g H s. Next, it is necessary to find the distinct representatives of the double cosets S n 1 \S n /. The set C = {(1), (1, n), (2, n),..., (n 1, n)} is a complete set of coset representatives for the right cosets S n 1 \S n. Now, for a given corresponding to the partition λ = (λ 1, λ 2,..., λ k ) of n, consider two representatives σ 1 and σ 2 from the same (S n 1, )-double coset. Then it must be the case that τ 2 σ 1 τ 1 = σ 2 for some τ 2 S n 1 and τ 1. Suppose that σ 1 (i) = n, and τ 1 (j) = i, which occurs if and only if i and j are in the same subset {( i 1 λ j ) + 1, ( i 1 λ j ) + 2,..., ( i 1 λ j ) + λ i } on which i acts. Since τ 2 S n 1, τ 2 fixes n, and hence τ 2 σ 1 τ 1 (j) = n, and so also σ 2 (j) = n. Hence, if i and j are in the same row of the Young tableau [ ], then (i, n) and (j, n) represent the same (S n 1, )-double coset. Also, if i and j are in different rows of [ ], then (i, n) and (j, n) represent distinct (S n 1, )-double cosets. Thus, a complete set of all (S n 1, )-double coset representatives is any subset {(i 1, n), (i 2, n),..., (i k, n)} of the set C such that each i l is in a different row of the Young tableau [ ]. Taking any σ = (i l, n) from the set of (S n 1, )-double coset representatives, the Young subgroup σ σ 1 gives a Young tableau which has the same shape as [ ] but which reverses the positions of i l and n. Intersecting this with S n 1 has the effect of removing from the Young tableau the last box of the row j containing n, since by the construction of the Young tableau, n is written in the last box of a collection of rows of the same size. This gives a Young tableau for S n 1 of shape γ = (λ 1, λ 2,..., λ j 1,..., λ k ). For each row of [ ] which has the same length as row j, conjugating by a transposition corresponding to an element in that row and intersecting with S n 1 gives a Young subgroup isomorphic to σ σ 1 intersected with S n 1, since the resulting Young tableau has the same shape γ. It then follows from Theorem 3.4 that each of the corresponding induced representations are isomorphic. So, for each partition γ λ, there are exactly c(λ, γ) copies of this induced representation. Thus, the direct sum is taken over all possible γ λ, where each of the respective induced representations is multiplied

23 4 REPRESENTATIONS OF THE SYMMETRIC GROUPS 23 by the factor c(λ, γ). 4.6 The Snapper Conjecture The mistake that Vershik makes occurs when he sets out to prove his third lemma, which can be stated as follows: Theorem 4.5. (Lemma 3 in [10]) The irreducible representations π µ appearing in the decomposition of the induced representation Ind Sn 1 correspond to partitions µ which dominate λ. This lemma is previously known to be true. The original formulation of this problem was put forth by Snapper, who proved that [ Ind Sn 1, π µ > 0] [µ λ] and conjectured, but did not prove, that the converse was also true (see [8]). Two years later, the converse was in fact shown to be true by Liebler and Vitale (see [3]). Because of its history, Theorem 4.5 is sometimes referred to as the Snapper conjecture. Despite this background knowledge, Vershik seeks a new inductive proof relying on combinatorics and the branching rules. However, in the translation of the representation theory into combinatorics, Vershik asserts an equivalence between two statements where there is actually only an implication in one direction Combinatorial Statement In Vershik s attempted proof of Theorem 4.5, he makes one central claim, namely, he gives the following two statements about the partitions λ n, µ n and asserts that they are equivalent: Statement 4.6. µ λ Statement 4.7. For all ρ (n 1) for which ρ µ, there exists γ (n 1) for which γ λ such that ρ γ. One of these two implications is true, that is: Theorem 4.8. Statement 4.6 implies Statement 4.7.

24 4 REPRESENTATIONS OF THE SYMMETRIC GROUPS 24 Proof. Define µ = (µ 1, µ 2,..., µ a ) and define λ = (λ 1, λ 2,..., λ b ). If µ λ, then by definition, for all 1 i max {a, b} i µ j i λ j. Now define the diagram [ρ] to be the diagram [µ] with one block removed from the end of row k. Using this definition, one may see that [λ] must also have at least k rows, otherwise k 1 k 1 µ j < n = λ j, which would violate the dominance ordering. Define a removable block to be a block of a diagram [λ] which may be removed to yield another valid diagram [γ] (such that γ λ). Every diagram has at least one removable block, since the rightmost cell of the bottom row of any diagram must be removable. Also, except for the removable block in the final row, there is a removable block in row i of [λ] if and only if row (i + 1) has length strictly less than row i. With this definition in hand, consider two cases. Case 1: Suppose that [λ] has a removable block in row l k and form the diagram [γ] by removing the removable block in row l of [λ]. Then: for m < l, µ m = ρ m and λ m = γ m, for l m < k, µ m = ρ m and ( for m k, ( µ j ) 1 = λ j ) 1 = γ j, ρ j and ( λ j ) 1 = γ j. Since for all i, i µ j i λ j, one can use these three equations to show that, for all i, i ρ j i γ j, which in turn implies that ρ γ as desired. Case 2: Now assume that [λ] has no removable block in any row r k. Define l to be the uppermost row of [λ] which has a removable block. Then it must be true that l > k. Also, for all 1 m l: λ j = mλ 1 µ j,

25 4 REPRESENTATIONS OF THE SYMMETRIC GROUPS 25 where the equality on the left is due to the fact that having no removable block in the first (l 1) k rows implies that the first l rows all have the same length, namely that of λ 1, and where the inequality on the right is due to the dominance ordering. Now, consider k µ j. Since [µ] has a removable block in row k, it follows that µ k > µ k+1. Also, since [λ] does not have a removable block in row k, it must be true that λ k = λ k+1. Combining these two facts with the dominance ordering on µ and λ, it must be true that k λ j < k µ j since equality would imply that k+1 λ j > k+1 µ j, in violation of the dominance ordering. Following the same line of reasoning used in Case 1, it may now been seen that for m < k, µ m = ρ m and λ m = γ m, for k m < l, ( µ j ) 1 = ρ j and λ m = γ m, for m l, ( µ j ) 1 = ρ j and ( λ j ) 1 = γ j. Hence, since [ k λ j < k µ j] [ k λ j ( k µ j) 1], these three statements together imply that ρ λ Combinatorial Counterexamples Despite Theorem 4.8, a relatively simple counterexample shows that the equivalence which Vershik asserts is in fact false. Consider the partitions µ = (2, 2, 2) and λ = (3, 1, 1, 1) of n = 6. Recall that, as noted in Section 2.3, these two partitions cannot be compared using the dominance ordering. [µ] =, [λ] =

26 4 REPRESENTATIONS OF THE SYMMETRIC GROUPS 26 For the given µ, the only possible ρ µ corresponds to the partition ρ = (2, 2, 1). There are two choices for γ, but choosing the partition γ = (2, 1, 1, 1) it is clear that ρ γ. Hence, Statement 4.6 is false while Statement 4.7 is true, and so the two statements are not equivalent. While this particular counterexample occurs when n = 6, there exist counterexamples for greater values of n as well. The value n = 6 yields the smallest possible counterexample, since, as mentioned in Section 2.3, the dominance ordering is a total ordering for n 5, so in these instances it would not be possible to find two partitions µ, λ such that µ λ and λ µ. Surprisingly enough, however, one can find an infinite set of counterexamples when n 6. For instance, the above argument can be extended to any even n 6 (specifically, define n = 2k for some k N) by defining the diagram [µ] to have k rows of length 2 and defining [λ] to be the diagram with one row of length 3 and (n 3) rows of length 1. Then µ λ, since 2 < 3. But now choose [γ] to be the diagram [λ] with one block removed from the uppermost row; [ρ] must be be the diagram [µ] with one block removed from the bottommost row. Hence, ρ γ, leading to the previously-established contradiction, and so as n ranges over the positive integers greater than or equal to 6, this construction can be used to create an infinite class of counterexamples Representation Theory Counterexamples With this background, one now can translate the combinatorial counterexamples given in Section into representation theoretic counterexamples. In what follows, define the value of the inner product of two representations to be the value of the inner product of the corresponding characters, as defined in Section 3.7. Recall that, as noted in the discussion following Theorem 4.5, Vershik attempts to use the branching rules to prove the known fact (see [8], [3]) that π µ, Ind Sm 1 0 µ λ, which is nothing more than a restatement of Theorem 4.5.

27 4 REPRESENTATIONS OF THE SYMMETRIC GROUPS 27 To begin, note that the implication π µ, Ind Sm 1 0 for all ρ µ, π µ, Ind Sm 1 0 (4) ρ µ is valid, since one may break the inner product of the direct sum into a sum of inner products, each of which are greater than or equal to zero. For at least one µ in this sum (specifically, µ = µ) the inner product is not equal to zero, hence it must be greater than zero, and thus the sum of the inner products must also be greater than zero. Then, using the branching rule for induced representations (Theorem 4.2), one can see that the right side of this implication is equivalent to the following statement: for all ρ µ, Ind Sm S m 1 π ρ, Ind Sm 1 0. By Frobenius Reciprocity (see Theorem 3.5), this statement is equivalent to the assertion below: for all ρ µ, π ρ, Res Sm 1 Ind Sm 1 0. Theorem 4.4 can now be applied to change the restriction of the induction of the character 1 into a direct sum of representations: for all ρ µ, π ρ, γ λ c(λ, γ)ind S m 1 S γ 1 0. Since an inner product is sesquilinear, the direct sum within the inner product can be rewritten as a sum of inner products. Additionally, since each c(λ, γ) is a positive integer, these coefficients may be pulled out of each inner product just as they are, without needing to be conjugated. To be sure that the sum of the resulting inner products is not equal to zero, there must be at least one term of the sum which is not equal to zero, hence: for all ρ µ, there exists a γ λ such that π ρ, Ind S m 1 S γ 1 0. Finally, applying the equivalence in Lemma 4.5 shown by Snapper, Liebler, and Vitale, the statement about inner products may be converted into a statement about partitions of (n 1): for all ρ µ, there exists a γ λ such that ρ γ.

28 5 FURTHER QUESTIONS 28 This is none other than Statement 4.7 from the section above. Now, Vershik s erroneous claim is that this statement is equivalent to Statement 4.6, but as shown in Section 4.6.2, this is false. Still, by Theorem 4.5, [ Ind Sn 1, π µ > 0] [µ λ] is true, as noted in the introduction to Section 4.6. All of the above equivalences and implications can be used to show that the converse of statement (4) is false. First, recall that Statement 4.7 does not imply Statement 4.6 due to the combinatorial counterexamples given in Section Now, following the above chain of equivalences from the bottom upwards, it must be the case that from Statement 4.7 implies the right side of statement (4). If the converse of statement (4) were true, then this chain of implications would continue: the right side of statement (4) would imply the left side, which in turn would imply Statement 4.6 by Snapper s proof. Therefore, were the converse of statement (4) true, then Statement 4.7 would imply Statement 4.6, which has already been shown to be false. So the converse of statement (4) must be false. In summary, for every combinatorial counterexample in which for all ρ µ there exists γ λ such that ρ γ, and yet µ λ, one can find that for all ρ µ, ρ µ π µ, Ind Sm 1 = 0, and yet the representation π µ does not appear in the decomposition of Ind Sm 1 into irreducible representations. 4.7 Frobenius-Young Correspondence, Revisited As noted in the introduction to this section, despite the error in Vershik s proof, the Frobenius-Young correspondence is still known to be true (see [1]). It may even be that there is a combinatorial proof of the correspondence which uses the branching rules, as Vershik attempted to do, but such a proof would require a different method of argument. Any valid combinatorial method would need to arrive at the intended result by a proof which can altogether avoid the equivalence Vershik tried, and failed, to show. 5 Further Questions The most obvious and important question which arises from the given considerations of Vershik s paper is whether there is indeed a a combinatorial proof of the Frobenius-Young correspondence using the branching rules. Another question to investigate would be whether other known theorems on the representation the-

29 REFERENCES 29 ory of S n, such as the Littlewood-Richardson Rule or the Murnaghan-Nakayama Rule (see Sections 4.9 and 4.10 in [6]), can be proven using the point of view established in Vershik and Okounkov s earlier papers ([5] and [9]). One would hope that answers to such questions could help to complete another proof of the Frobenius-Young correspondence and to continue to broaden the perspective of the symmetric groups as pioneered by Vershik and Okounkov. References [1] W. Fulton and J. Harris, Representation Theory: A First Course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, [2] G. James and M. Liebeck, Representations and Characters of Groups. Second edition. Cambridge University Press, New York, [3] R.A. Liebler and M.R. Vitale, Ordering the Partition Characters of the Symmetric Group, J. of Algebra 25 (1973), [4] I.G. Macdonald, Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, [5] A. Okounkov and A.M. Vershik, A New Approach to Representation Theory of Symmetric Groups, Selecta Math. (N.S.) 2 (1996), no. 4, [6] B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Second edition. Graduate Texts in Mathematics, 203. Springer-Verlag, New York, [7] J.-P. Serre, Linear Representations of Finite Groups. Translated from the second French edition by Leonard L. Scott. Graduate Texts in Mathematics, 42. Springer-Verlag, New York-Heidelberg, [8] E. Snapper, Group characters and nonnegative integral matrices, J. of Algebra 19 (1971), [9] A.M. Vershik and A. Okounkov, A New Approach to Representation Theory of Symmetric Groups, II. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 307 (2004),

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