YOUNG-JUCYS-MURPHY ELEMENTS MICAH GAY Abstract. In this lecture, we will be considering the branching multigraph of irreducible representations of the S n, although the morals of the arguments are applicable to more general cases. We will liberally apply criteria about the centralizer of a subrepresentation to show that that the restriction of an irreducible representation to a subrepresentation has simple multiplicity, which will show that the branching graph of irreducible representations of S n is in fact simple. We will then define the Young-Jucys-Murphy elements in C[S n], show that they in fact generate the Gelfand-Tsetlin algebra, and see how they relate to the GZ-basis. N.B. We will assume that all vector spaces are finite-dimensional over C. Contents 1. Set-up 1 2. Assorted Lemmas, Tidbits, and Facts 2 3. Centralizers and Centres 4 4. Young-Jucys-Murphy Elements 4 5. Takeaways 6 References 6 1. Set-up Recall the following: Let {1} = G 0 < G 1 < G 2 < be a chain of finite groups, and denote Ĝ = {irreps of G} (irreducible = no G-invariant subspaces). For each ρ Ĝn, V ρ decomposes into a direct sum of irreps µ Ĝn 1 with multiplicities m µ = dim Hom G (V µ, V ρ ). V ρ = (V µ ) mµ. µ Ĝn 1 The branching graph is the directed multigraph whose vertices are elements of k 0 Ĝk, with Ĝn called the nth level. Two vertices µ Ĝn 1 and ρ Ĝn are connected by k dirrected edges if k = dim Hom Gn 1 (V µ, V ρ ). 1
2 MICAH GAY Write µ ρ if µ and ρ are connected, i.e if V µ is a factor in the decomposition of V ρ. Call the branching graph simple if all multiplicities are either 0 or 1, in which case V ρ = T V t V t := C, t T = { increasing paths t = {ρ 0 ρ 1 ρ n = ρ}, ρ i Ĝi } Choose units v t V t t T, then the GZ basis of V ρ := {v t : t T } Remark 1.1. What this looks like is choosing some generator of V {1}, and tracking its image under each path in V ρ. Definition 1.2. The Gelfand-Tsetlin algebra GZ n is the algebra generated by the centers Z 1 C[G 1 ],..., Z n C[G n ]; that is, GZ n =< Z 1,, Z n > Proposition 1.3. GZ n is the maximal commutative subalgebra in C[G n ] when the branching graph is simple, and consists of operators that are diagonal in the GZ-basis. Remark 1.4. We will eventually be looking at how GZ n acts on each irrep. We also recall the following lemma which will be very handy: Lemma 1.5. ( Criteria ): Let M be a semisimple finite-dimensional C-algebra, N M a subalgebra. The centralizer Z N (M) is commutative if and only if, for any ρ M, the restriction Res M N V ρ of an irrep of M to N has simple multiplicities. As well, keep in mind that irreps (i.e. irreducible representations) of a finite group G C[G] modules. N.B. From now on, take G n = S n. 2. Assorted Lemmas, Tidbits, and Facts We will be looking at the branching graph of irreps of C[S n ] to study the branching graph of S n. We can do this because we have a bijection: Hom groups (G, GL(n, C)) = Hom unital (C[G], Mat nxn (n, C)) assoc. algebras To see why this is, if you have any homomorphism on the elements of G, you can extend it to C[G], and likewise any homomorphism of C[G] can be restricted to the basis G. We also need to think about invertibility, but will hand-wave that for now. While S n has a trivial centre, the group algebrac[s n ] does not, so we can get more information by working with it. Note the following fact from elementary algebra: Fact 2.1. Conjugation preserves cycle type, and if σ = (i 1,, i k ), then τστ 1 = (τ(i 1 ),, τ(i k )) Proof. Consult any algebra textbook ever written.
YOUNG-JUCYS-MURPHY ELEMENTS 3 As we will be proving results about the centres and centralizers of C[S n ], let s think about what the centre Z n := Z(C[S n ]) = { z C[S n ]) : zy = yz y C[S n ]) } looks like: Proposition 2.2. Let z Z n, z = g S n c g g. For any g S n, h S n, hzh 1 = z. Since conjugation will permute the g i s within the conjugacy class [g] (which consists of all permutations of a particular cycle type), we must have that c gi s are all the same, i.e. z = c [g] g i [g] g i [g] So, we can describe Z n as: where Z n = {Z λ λ n} Z λ = σ S n w/cycletypeλ Lemma 2.3. (Lemma 1) Every g S n is conjugate to g 1 S n, i.e. h S n s.t. g 1 = hgh 1. Moreover, we can find such an h S n 1. σ Proof. Clearly every permutation S n is conjugate to its inverse, since if σ = (i 1,..., i n ), then σ 1 = (i n, i n 1..., i 1 ), ρσρ 1 = (ρ(i 1 ),...ρ(i n )), choose ρ s.t. ρ(i 1,..., i n ) = (i n, i n 1,..., i 1 ). For g S n, let g S n 1 be the induced permutation in S n 1. Take h S n 1 that conjugates g and g 1, so g 1 = hg h 1. Then h has the fixed point n, so extended to S n, it satisfies g 1 = hgh 1. Before the next lemma, it s time for another definition: Definition 2.4. An involution algebra, or a algebra, is an algebra A with a map (called an involution) : A A satisfying: (1) (a ) = a (2) (ab) = b a (3) (λa + b) = λa + b Call a the conjugate or adjoint of a. Example 2.5. C is a -algebra over R with = conjugation. Definition 2.6. In an algebra A, call a A normal if a commutes with its conjugate, i.e. aa = a a, and call a self adjoint if a = a. Lemma 2.7. (Lemma 2) Let A be an algebra over C. Then (1) A is commutative all of its elements are normal. (2) If every real element is self-adjoint, then A commutative. Proof. (1) : Trivial : Suppose aa = a a a A, and denote A sa = {a A : a self-adjoint}. A can be decomposed as A = A sa + ia sa (by properties of ). If a, b A sa, then a = a and b = b, so (a + ib) = a + īb = a ib. But (a + ib) normal (a + ib)(a ib) = (a ib)(a + ib) ab = ba, i.e. a and b commute. Hence A commutative.
4 MICAH GAY (2) Let A R be the real subalgebra of A, i.e. A = C A R, and assume all a A are self-conjugate (i.e. a = a A A). Then a, b A R ab = (ab) = b a = ba, so A R commutative, but then so is A = C A R. 3. Centralizers and Centres Theorem 3.1. The centralizer Z C[Sn 1](C[S n ]) =: Z n 1 (n) of C[S n 1 ] in C[S n ] is commutative. Proof. By lemma 2, we have reduced this problem to checking that every real element of the centralizer Z n 1 (n) C[S n ] is self-adjoint. Let { g i : g i S n } be a basis for C[S n ], f = i c ig i Z n 1 (n) c i s R. What does self-adjoint look like in C[S n ]? The adjoint of g, viewing g as a function g : C[S n ] C[S n ], can also be categorized as the function g : C[S n ] C[S n ] such that< ga, b >=< a, g b > a, b C[S n ]. Take the inner product < g i, g j >= δ ij. Then < g i, g j >=< gg i, gg j >, and by def. of adjoint we have < gg 1, g 2 >=< g 1, g g 2 >=< g 1, g 1 g 2 > g = g 1. So, ( i c ig i ) = i c i g 1 i for f is unique, this expression is invariant under conjugation by h S n 1, f hfh 1 = f. By lemma 1, we can choose h i s.t. h i g i h 1 i summand c i g 1 = i c ig 1 i. f Z n 1 (n) commutes with all h S n 1, so since the expression, so i c ig i i c ig 1 i. But since hfh 1 = f, that means that c i g i a i a summand, so f = f. = g 1 i In light of this, we have the following: Theorem 3.2. A a finite-dimensional -algebra over R, B a -subalgebra. Let G = { g i } be a basis of A that is closed under, and s.t. i, orthogonal b i B (b = b 1 ) s.t. b i g i b i = gi. Then Z B(A) is commutative, and hence the restriction of any irrep from A to B has simple multiplicity. In particular, if A is a group algebra of a finite group G and B of H G, where A has basis G, then g G, h H, g G s.t. h 1 g h = g, and if we can take g = g, then Z B (A) is commutative, and hence the restriction of any irrep from A to B has simple multiplicity. Remark 3.3. The proof we just gave for the simplicity of the spectrum (i.e. the branching of the multigraph) didn t require any knowledge of the representations themselves, only elementary algebraic properties of the group. 4. Young-Jucys-Murphy Elements Remark 4.1. We can also look at the branching through analyzing the centralizers of the group algebras, so we re going to develop a more detailed description of the centralizer Z n 1 (n) and its relation to GZ n. Definition 4.2. The Y oung Jucys Murphy elements (henceforth abbreviated as Y JM elements) are X i = (1i) + (2i) +... + ((i 1)i) C[S n ] Remark 4.3. X i = (transpositions in S i ) - (transpositions in S i 1 ), so X i is the difference of an element in Z(i) and Z(i 1), so X i GZ n ; in particular, the X i s commute. Proposition 4.4. X k / Z k for any k.
Proof. Z k = span{ σ S n cycletypeλ YOUNG-JUCYS-MURPHY ELEMENTS 5 σ λ a partition of k }, so for cycle type λ = 2,1,1,...,1, any element Z n consisting of 2-cycles must be expressed in terms of all the 2-cycles S n. Theorem 4.5. The centre Z n C[S n ] is a subalgebra of the one generated by the centre Z n 1 C[S n 1 ] and X n : Z n < Z n 1, X n > Proof. (Sketch) (1) Show all classes of one-cycle type permutations lie in < Z n 1, X n >: X n = n 1 i=1 (in) = n i j;i,j=1 (ij) - n 1 i j;i,j=1 (ij), where the first sum < Z n 1, X n > and the second Z n 1. Note that Xn 2 = n i j;i,j=1 (in)(jn) = n 1 i j;i,j=1 (ijn)+1, so X2 n+ n 1 i j k;i,j,k=1 (ijk) = n i j k;i,j,k=1 (ijk) < Z n 1, X n >. In a similar fashion, we can show that the sum Xn k 1 + n i 1 i 2... i k ;i 1,...,i k =1 (i 1 i k ) < Z n 1, X n >, so that all one-cycle type classes in Z n lie in < Z n 1, X n >. (2) Apply the general theorem that Z n is generated by the classes of one-cycle type permutations (i.e. these permutations are the multiplicative generators) to conclude that Z n < Z n 1, X n >. Corollary 4.6. The algebra GZ n is generated by the Y JM elements, i.e. GZ n =< X 1,...X n > Proof. By definition, GZ n =< Z 1,..., Z n >. We will proceed by induction: GZ 2 = C[S 2 ] =< X 1 = 0, X 2 >= C. Now, assume by induction that GZ n 1 =< X 1,..., X n 1 >. NTS GZ n =< GZ n 1, X n >. : Clearly GZ n < GZ n 1, X n > since GZ n 1 GZ n and X n Z n. : By the previous theorem, Z n < Z n 1, X n > < GZ n 1, X n >. Proposition 4.7. X k / Z k for any k. Proof. Z k = span{ σ S n cycletypeλ 2-cycles must be expressed in terms of all the 2-cycles S n. σ λ k }, so for cycle type λ = 2,1,1,...,1, any element Z n consisting of Theorem 4.8. The centralizer Z n 1 (n) is generated by the centre Z n 1 C[S n ] and the YJM-element X n : Z n 1 (n) =< Z n 1, X n > Proof. Let s consider a basis for Z n 1 (n). This will be the union of the basis of Z n 1, along with classes of the form: { (i 1,1 i 1,k 1 n)(i 2,1 i 2,k2 ) (i j,1 i n,kj ) k 1,..., k j n} where i j,k { 1,..., n-1 } and the cycle lengths k 1,..., k j over possible partitions of n. This is because conjugating by any element h Z n 1 must fix any z Z n 1 (n), and so h( (i 1,1 i 1,k1 1n)(i 2,1 i 2,k2 ) (i n,1 i n,kn ))h 1 = (h(i 1,1 ) h(i 1,k 1 )h(n))(h(i 2,1 ) h(i 2,k2 )) (h(i n,1 ) h(i n,kn )) = (h(i 1,1 ) h(i 1,k 1 )n)(h(i 2,1 ) h(i 2,k2 )) (h(i n,1 ) h(i n,kn )) = (i 1,1 i 1,k 1 n)(i 2,1 i 2,k2 ) (i n,1 i n,kn )
6 MICAH GAY Since n must stay in the cycle of length k 1 and h permutes the rest of the elements. As in the proof showing that Z n < Z n 1, X n >, we can take the sum of these classes with the corresponding classes: { (i 1,1 i 1,k )(i 2,1 i 2,k2 ) (i j,1 i n,kj ) k 1,..., k j n} from Z n 1 to obtain classes in Z n, which shows that a basis of Z n 1 can be obtained via a linear combination of elements in the bases of Z n 1 and Z n : But we already proved that Z n < Z n 1, X n >, so Z n 1 (n) < Z n 1, Z n > Z n 1 (n) < Z n 1, X n > Conversely, Z n 1 Z n 1 (n) and X n commutes with all elements Z n 1, so we also have Z n 1 (n) < Z n 1, X n > 5. Simplicity of Branching Theorem 5.1. (Main) The branching of the chain C[S 1 ] C[S n ] is simple, hence the same is true for S 1 <... < S n. Proof. Z n 1 (n) < Z n 1, X n > GZ n commutative any restriction from C[S n ] to C[S n 1 ] is simple by Criteria. Corollary 5.2. GZ n is the maximal commutative subalgebra of C[S n ], and in each irrep of S n, the GZ-basis is determined up to scalar factors (as a consequence of Schur s lemma). Definition 5.3. The union of GZ-bases of irreps ˆ S n is called the Y oung basis. Proposition 5.4. Let v be a vector in the Young basis of some irrep, and denote the weight of v: α(v) = (a 1,..., a n ) C n be the eigenvalues of X 1,..., X n on v. Denote the spectrum of YJM-elements as Spec(n) = {α(v) v Y oungbasis} Then α Spec(n) determines v up to scalar multiplication, and we can easily see that Spec(n) = λ ˆ S n dim(λ) = dim(gz n ) i.e. dim(gz n ) = the sum of pairwise non-isomorphic irreps Proposition 5.5. We also have a bijection between Spec(n) and the set of paths in the branching graph, and a natural equivalence relation on Spec(n): if v α and v β the corresponding vectors in the Young basis corresponding to weights α and β Spec(n), then α β if the paths T α and T β have the same end, i.e. v α and v β are in the same irrep, so that Spec(n)/ = S ˆ n Remark 5.6. Later, we will see how these eigenvectors can be found from combinatorial data from Young tableaux. 6. Takeaways (1) YJM-elements X n = (1n) +... + ((n 1)n) (2) GZ n =< Z 1,..., Z n >=< X 1,..., X n > (3) The branching of the multigraph of C[S n ] and therefore that of S n is simple (4) The spectrum of the YJM-elements determines the branching of the graph
YOUNG-JUCYS-MURPHY ELEMENTS 7 References [1] Vershik and Okounkov A New Approach to the Representation Theory of Symmetric Groups, II ftp://ftp.ams.org/pub/tex/doc/amsmath/short-math-guide.pdf