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READING REPORT Symmetric Jordan Basis, Terwilliger Algebra of Binary Hamming Scheme and Gelfand-Tsetlin Basis

Yizhe Zhu

Shanghai Jiao Tong University zyzwstc@sjtu.edu.cn

December 13, 2014

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 1 / 41

Overview

1

Symmetric Jordan Basis a Linear Analog of SCD the Linear BTK Algorithm SJB of V (B(n))

2

Terwilliger algebra of binary Hamming scheme Association Schemes Block Diagonalization

3

Gelfand-Tsetlin basis Branching Graph Gelfand-Tsetlin Basis Gelfand-Tsetlin Diagrams

4

Remark

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 2 / 41

Symmetric Chain Decomposition (SCD)

Definition (SCD)

Let P be a finite graded poset with n = r(P),where r : P → N is the rank function.If p ≤ q, we have r(p) ≤ r(q). We say p covers q in P if r(p) = r(q) + 1. The rank of P is r(P) = max{r(p) : p ∈ P}. Let Pi denote the set of elements in P with rank i. A symmetric chain of a graded poset P is a sequence (p1, ..., ph) of elements in P such that pi covers pi−1 for i = 2, ..., h, and r(p1) + r(ph) = r(P), if h ≥ 2 or 2r(p1) = r(P), if h = 1. A symmetric chain decomposition (SCD) of P is a decomposition of P into pairwise disjoint symmetric chains.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 3 / 41

Symmetric Chain Decomposition (SCD)

Example (SCD of B(4), Greene - Kleitman)

Figure: the symmetric chain decomposition of B(4)

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 4 / 41

Symmetric Jordan Basis

Definition (SJB)

P is defined as before, let V (P) denote the complex vector space with P as basis. Then we have V (P) = V (P0) ⊕ V (P1) ⊕ ... ⊕ V (Pn). An element v ∈ V (P) is homogeneous if v ∈ V (Pi) for some i. A linear map U : V (P) → V (P) is said to be up operator if for all p ∈ P, U(p) =

q q is the sum of the all elements covering p. We define

U(p) = 0 if p is a maximal element of P. A Jordan chain in V (P) is a sequence v = (v1, ..., vh) of nonzero homogeneous elements such that U(vi−1) = vi for i = 2, ..., h and U(vh) = 0. We say v is symmetric, if r(v1) + r(vh) = r(P) for h ≥ 2 or 2r(v1) = r(P) for h = 1. A symmetric Jordan basis (SJB) of V (P) is a basis of V (P) consisting of a disjoint union of symmetric Jordan chains in V (P).

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 5 / 41

Symmetric Jordan Basis

Example

The SJB of V (B(2)) is given by two chains: ((0, 0), (1, 0) + (0, 1), 2(1, 1)), ((0, 1) − (1, 0)). Because r((0, 0)) + r(2(1, 1)) = 2, 2r((0, 1) − (1, 0)) = 2, U((0, 0)) = (1, 0) + (0, 1), U((1, 0) + (0, 1)) = (1, 1) + (1, 1) = 2(1, 1) and U((0, 1) − (1, 0)) = (0, 0).

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 6 / 41

The Linear BTK Algorithm

Let k1, ...kn be nonnegative integers. Define the poset M(n, k1, ..., kn) = {(x1, ..., xn) ∈ Nn : 0 ≤ xi ≤ ki, for all i} with partial

• rder defined by componentwise ≤. When k1 = k2 =, , , = kn we write

M(n, k) for M(n, k..., k). Moreover, when k = 1, it is Boolean algebra B(n). An algorithm to construct an explicit SCD of M(n, k1, ..., kn) was given by de Bruijin, Tengbergen and Kruyswijk, called BTK. Here we present a linear analog of BTK algorithm.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 7 / 41

The Linear BTK Algorithm

The basic bulding block of the linear BTK is an inductive method for constructing a SJB of V (M(2, p, q)).

Lemma

Let p, q be positive and set P = M(2, p, q), W = V (P) with up operator

• U. dimW = (p + 1)(q + 1). The action of U on the standard basis of W

is given as follows: for 0 ≤ i ≤ p, 0 ≤ j ≤ q U((i, j)) =        (i + 1, j) + (i, j + 1) if i < p, j < q (i + 1, j) if i < p, j = q (i, j + 1) if i = p, j < q if i = p, j = q

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 8 / 41

The Linear BTK Algorithm

Let v = (v(0), v(1), v(2), ..., v(p + q)) be the symmetric Jordan chain in W generated by v(0) = (0, 0). Then v(k) =

i,j

k

i

• (i, j), 0 ≤ k ≤ p + q.

Define the homogeneous vector in W = V (M(2, p, q)) as follows. v(i, j) = (p − i)(i, j) − (q − j + 1)(i + 1, j − 1), 0 ≤ i ≤ p − 1, 1 ≤ j ≤ q.

Theorem

(1) {v(k)|o ≤ k ≤ p + q} ∪ {v(i, j)|0 ≤ j ≤ p − 1, 1 ≤ j ≤ q} is a basis of W . (2) For 0 ≤ i ≤ p − 1, 1 ≤ j ≤ q we have U(v(i, j)) =        v(i + 1, j) + v(i, j + 1) if i < p − 1, j < q v(i + 1, j) if i < p − 1, j = q v(i, j + 1) if i = p − 1, j < q if i = p − 1, j = q

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 9 / 41

The Linear BTK Algorithm

Thus the action of U on the v(i, j) is isomorphic to the action of U on the standard basis (i, j) of V (M(2, p − 1, q − 1)), except that the map (i, j) → v(i, j + 1), (i, j) ∈ M(2, p − q, q − 1) shifts ranks by one. We use the following example to show the induction works.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 10 / 41

Example (SJB of V (M(2, 2, 2)))

The SJB of V (M(2, 1, 1)) is given by two chains: ((0, 0), (1, 0) + (0, 1), 2(1, 1)) ((0, 1) − (1, 0)). Transferring these chains to V (M(2, 2, 2)) via the map (i, j) → v(i, j + 1) gives the following chains. (v(0, 1), v(1, 1) + v(0, 2), 2v(1, 2)) (I) (v(0, 2) − v(1, 1)) (II) where v(i, j) are given by v(0, 1) = 2(0, 1) − 2(1, 0) v(1, 1) = (1, 1) − 2(2, 0) v(0, 2) = 2(0, 2) − (1, 1) v(1, 2) = (1, 2) − (2, 1) The symmetric Jordan chain generated by v(0) = (0, 0) is given by ((0, 0), (1, 0) + (0, 1), (2, 0) + 2(1, 1) + (0, 2), 3(2, 1) + 3(1, 2), 6(2, 2)) (III) Chains (I), (II), (III) form a SJB of V (M(2, 2, 2))

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 11 / 41

Induction

Let BTK(2, p, q) be the set of SJB of V (M(2, p, q)) produced by the linear BTK algorithm. Now we can summarize the induction from BTK(2, p − 1, q − 1) to BTK(2, p, q) as follows. 1 For any (i, j) ∈ BTK(2, p − 1, q − 1), the map (i, j) → v(i, j + 1) creates the elements in BTK(2, p, q) and keeps the symmetric Jordan chain. 2 Starting from v(0) = (0, 0) ,the up operator U creates a symmetric Jordan chain v = (v(0), v(1), v(2), ..., v(p + q)). 3 The symmetric Jordan chains created by Step 1 and Step 2 form the BTK(2, p, q). The case V = V (M(n, k1, ..., kn)), n ≥ 3 can be reduced to the case n = 2 by induction. Then we complete the linear BTK algorithm.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 12 / 41

SJB of V (B(n))

When applied to V (B(n)), the linear BTK algorithm can produce an SJB with very interesting properties. Let , denote the standard inner product on V (B(n)), i.e.,X, Y = δ(X, Y ), (Kronecker delta) for X, Y ∈ B(n). The length

• v, v of v ∈ V (B(n)) is denoted v .

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 13 / 41

SJB of V (B(n))

Proposition

Let O(n) be the SJB produced by linear BTK when appled to B(n). (1) The elements of O(n) are orthogonal wrt the standard inner product. (2) Let 0 ≤ k ≤ [n/2] and let (xk, ..., xn−k) and (yk, ..., yn−k) be any two symmetric Jordan chains in O(n) starting at rank k and ending at rank n − k. Then xu+1

xu = yu+1 yu , k ≤ u < n − k.

(3) In the notation of (2), we have, for k ≤ u < u − k,

xu+1 xu =

• (u + 1 − k)(n − k − u) = (n − k − u)

n−2k

u−k

1

2 n−2k

u+1−k

− 1

2 Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 14 / 41

Example (SJB of V(B(n)))

(1) The SJB of V (B(2)) is given by ((0, 0), (1, 0) + (0, 1), 2(1, 1)) ((0, 1) − (1, 0)) (2) The SJB of V (B(3)) is given by ((0, 0, 0), (1, 0, 0) + (0, 1, 0) + (0, 0, 1), 2((1, 1, 0) + (1, 0, 1) + (0, 1, 1)), 6(1, 1, 1)) (2(0, 0, 1) − (1, 0, 0) − (0, 1, 0), (1, 0, 1) + (0, 1, 1) − 2(1, 1, 0)) ((0, 1, 0) − (1, 0, 0), (0, 1, 1) − (1, 0, 1))

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 15 / 41

SJB of V (B(n))

More interesting properties can be said about SJB of V (B(n)) from Terwilliger algebra and Gelfand-Tsetlin basis.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 16 / 41

Symmetric Association Schemes

Definition (symmetric association schemes)

A symmetric association scheme of class d is a pair Y = (X, {Ri}d

i=0)

consisting of a finite set X amd relations R0, R1, ..., Rd on X such that: {Ri}d

i=0 is a partition of X × X.

R0 = {(x, x) : x ∈ X}. Ri = Rt

i for 0 ≤ i ≤ d, where Rt i = {(y, x) : (x, y) ∈ Ri}.

Given (x, y) ∈ Rh, ph

i,j = |{z ∈ X : (x, z) ∈ Ri, (z, y) ∈ Rj}| depends

• nly on h, i and j.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 17 / 41

Definition (Bose-Mesner algebra)

Given a symmetric association scheme Y = (X, {Ri}d

i=0), we define the

corresponding Bose-Mesner algebra as follows. For 0 ≤ i ≤ d, we define a matrix Ai by the formula. (Ai)x,y = 1 if (x, y) ∈ Ri

• therwise

We have A0 = I and AiAj = d

h=0 ph i,jAh. So A0, A1, ..., Ad form a basis

for a commutative algebra A, known as Bose-Mesner algebra. We write A1 as A.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 18 / 41

Definition (Dual Bose-Mesner algebra)

Given a symmetric association scheme Y = (X, {Ri}d

i=0), we define the

corresponding dual Bose-Mesner algebra as follows. Fix a vertex x ∈ X. For 0 ≤ i ≤ d, we define a diagonal matrix E ∗

i by the formula.

(E ∗

i )y,y =

1 if (x, y) ∈ Ri

• therwise

We have d

i=0 E ∗ i = I and E ∗ i E ∗ j = δi,jEi. So E ∗ 0 , E ∗ 1 , ..., E ∗ d form a basis

for a commutative algebra A∗, known as dual Bose-Mesner algebra.

Definition (Terwilliger algebra)

Let T denote the algebra generated by A and A∗. The algebra T is known as Terwilliger algebra.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 19 / 41

Definition (binary Hamming scheme)

Let X = {0, 1}d. Given vertices y = (y1, ..., yd) and z = (z1, ..., zd) in X we difine the Hamming distance: ∂(y, z) = |{i : yi = zi, 1 ≤ i ≤ d}| Then for 0 ≤ i ≤ d we define in X × X the following relation: Ri = {(x, y) ∈ X × X : ∂(x, y) = i} And ph

i,j = |{z ∈ X|∂(x, z) = i, ∂(z, y) = j}|.

The configuration (X, {Ri}d

i=0) is a symmetric association scheme, known

as the binary Hamming scheme H(d, 2). Remark: Such a scheme satisfies the P-polynomial property: given 0 ≤ i ≤ d and the matrix Ai, there exists a polynomial pi of degree i such that Ai = pi(A).

Proposition

The Terwilliger algebra of binary Hamming scheme H(d, 2) is generated by A and E ∗

0 , E ∗ 1 ..., E ∗ d

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 20 / 41

Block Diagonalization by SJB

Lemma

Let f : V (B(n)) → V (B(n)) be a linear map. Then f is Sn-linear if and

• nly if

Mf (X, Y ) = Mf (π(X), π(Y )), for all X, Y ∈ B(n), π ∈ Sn If we define An to be the set of all B(n) × B(n) complex matrices M satisfying M(X, Y ) = M(π(X), π(Y )), for all X, Y ∈ B(n), π ∈ Sn. By this lemma, we know An is isomorphic to EndSn(V (B(n))).

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 21 / 41

Block Diagonalization by SJB

Proposition

The Terwilliger algebra of H(n, 2), denoted by Tn, is isomorphic to the commutant of the Sn action on B(n). i.e., Tn = EndSn(V (B(n))) = An

Proof.

(1) Fix x = (0, ..., 0),then A, E ∗

0 , E ∗ 1 ..., E ∗ n ∈ An.

(2) E ∗

h AiE ∗ j = 0 if and only if ph i,j = 0.

The nonzero matrices of E ∗

h AiE ∗ j form a basis of An. Let |X| denote the

number of nonzero coordinates of X. (E ∗

h AiE ∗ j )(X, Y ) =

1 if |X| = h, |Y | = j, ∂(X, Y ) = i

• therwise

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 22 / 41

Block Diagonalization by SJB

Tn is a finite dimensional C ∗-algebra. Follows from C ∗-algebra theory that there exists a block diagonalization of Tn. Then there exists a B(n) × S unitary matrix N(n), for some index set S of cardinality 2n, such that N(n)∗TnN(n) is equal to the set of all S × S block-diagonal matrices      C0 . . . C1 . . . ... . . . Cm      , Ck =      Bk . . . Bk . . . ... . . . Bk      , where each Ck is block-diagonal matrix with qk identical blocks of order pk. Thus p2

0 + ... + p2 m = dim(Tn) and p0q0 + ... + pmqm = 2n.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 23 / 41

Block Diagonalization by SJB

By dropping duplicate blocks and keep one copy of B1, ..., Bm, we get a C ∗-algebra isomorphism.

Theorem

By the O(n) produced by linear BTK algorithm, we can explicitly determine the following block diagonalization. Φ : Tn ∼ =

m

• k=0

Mat(pk × pk) By the orthogonality of SJB, we can put each normalized basis as a column vector and form a unitary matrix. The main problem is how to put these vectors in the right order to construct the unitary matrix we want.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 24 / 41

Proof: (1) Construct the B(n) × S unitary matrix N(n)

For 0 ≤ k ≤ m, suppose O(n) contain qk symmetric Jordan chains, each containing pk vectors, starting at rank k and ending at rank n − k. Define the set S = {(k, b, i)|0 ≤ k ≤ m, 1 ≤ b ≤ qk, k ≤ i ≤ n − k}. For each k, fix some linear ordering of the qk Jordan chains from rank k to rank n − k. Then there is a bijection B : O(n) → S defined as follows: For any v ∈ O(n), define B(v) = (k, b, i), where i = r(v) and v occurs on the bth symmetric Jordan chain from rank k to rank n − k. Define a linear order on set S: (k, b, i) < (k′, b′, l′) iff k < k′ or k = k′, b < b′ or k = k′, b = b′, i < i′. Now we can form a B(n) × S unitary matrix N(n) as follows: The columns of N(n) are the normalized images

B−1(s) ||B−1(s)||, listed in

increasing linear order of S.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 25 / 41

Proof:(2) Determine the image of Φ

The nonzero images Φ(E ∗

h AiE ∗ j ) can be explicitly determined by some

calculations.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 26 / 41

Gelfand-Tsetlin Basis

We can also give a representiation theoretic characterization of this basis that explains its orthogonality. Namely, SJB is the symmetric Gelfand-Tsetlin basis.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 27 / 41

Representation Theory of Finite Groups

Recall some definitions of the representation theory of finite groups. A representation of a group G is a group homomorphism ρ : G → GL(n, C) from G to the general linear group GL(n, C). Let V and W be two vector spaces. (λ, V ) and (µ, W ) are two representations of a group G. If T ∈ Hom(V , W ) satisfying Tλ(g) = µ(g)T for all g ∈ G, we say T is an interwining operator. We denote by HomG(V λ, W µ) the vector space of all operators that interwine λ and µ. Two representations (λ, V ) and (µ, W ) are said to be equivalent, if there exists T ∈ HomG(V λ, W µ) which is bijective.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 28 / 41

Branching Graph

Definition (branching graph)

Let {1} = G(0) ⊂ G(1) ⊂ G(2) ⊂ ... be a chain of finite groups. G(n) ˆdenotes the set of equivalence classes of irreducible complex representation of the group G(n). The branching graph of the finite groups chain is defined by the following oriented graph. the vertices of the branching graph are the elements of the set

• n≥0 G(n)

ˆ two vertices µ ∈ G(n − 1) ˆand λ ∈ G(n) ˆare joined by k oriented edges if k = dimHomG(n−1)(V µ, V λ). Here k is the multiplicity of µ in the restriction of representation λ to group G(n − 1). We call G(n) ˆthe n th level of the branching graph, and write µ ր λ if µ and λ are connected by an edge.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 29 / 41

Gelfand-Tsetlin basis

From now on, we assume the finite groups sequence is the symmetric groups sequence {1} = S0 ⊂ S1 ⊂ S2 ⊂ ..., and denote the unique element in S0 ˆby φ.

Lemma

In the case G(n) = Sn, we always have k ∈ {0, 1}, which means the branching graph is multiplicity free. Now we fix an equivalence class of complex irreducible representation λ ∈ Sn ˆ , consider the decomposition V λ =

• µ∈Sn−1

ˆ ,µրλ

V µ which decomposes the irreducible Sn module into the sum of irreducible Sn−1 modules.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 30 / 41

Gelfand-Tsetlin basis

By induction, we obtain a canonical decomposition of the module V λ into irreducible S0 modules (1-dimensional subspaces). V λ =

• T

VT indexed by all possible paths T = λ0 ր λ1 ր ... ր λn, where λi ∈ Siˆand λn = λ, λ0 = φ.

Definition (Gelfand-Tsetlin (GZ) basis)

Choose a vector vT ∈ VT such that vT, vT = 1, where the inner product is the Sn invariant inner product in V λ. The basis {vT} is called the Gelfand-Tsetlin (GZ) basis of V λ.

Proposition

The Gelfand-Tsetlin basis is orthogonal with respect to the Sn invariant inner product in V λ.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 31 / 41

Gelfand-Tsetlin basis

A GZ vector v ∈

λ∈Sn ˆV λ is an element of the GZ basis (up to scalar) of

V λ for some λ ∈ Sn ˆ . For 1 ≤ i ≤ n, define Xi = (1, i) + (2, i) + ... + (i − 1, i) ∈ C[Sn]. The Xi’s are called the Young-Jucys-Murphy (YJM) elements. They generate the maximal commutaive subalgebra GZn of C[Sn].

Lemma

GZ vectors are the only vectors that are simultaneous eigenvectors of the action of X1, ..., Xn. Namely, for any GZ vector v, there exists (a1, ..., an) ∈ C such that Xiv = aiv for all 1 ≤ i ≤ n. We call α(v) = (a1, ..., an) the weight of v.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 32 / 41

Proposition

The SJB O(n), produced by the linear BTK algorithm when applied to B(n), is the Gelfand-Tsetlin vectors of V (B(n)) with the following properties: (1) Elements in each symmetric Jordan chain have the same weight. (2) (Conjecture) Elements in different symmetric Jordan chains have different weights. Remark: (1) is obvious because U is Sn-linear. (2) is true for n ≤ 4.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 33 / 41

Example (Elements in O(3) are Gelfand-Tsetlin vectors)

((0, 0, 0), (1, 0, 0) + (0, 1, 0) + (0, 0, 1), 2((1, 1, 0) + (1, 0, 1) + (0, 1, 1)), 6(1, 1, 1)) (I) (2(0, 0, 1) − (1, 0, 0) − (0, 1, 0), (1, 0, 1) + (0, 1, 1) − 2(1, 1, 0)) (II) ((0, 1, 0) − (1, 0, 0), (0, 1, 1) − (1, 0, 1)) (III) In this case, X1 = Id, X2 = (1, 2), X3 = (1, 3) + (2, 3) The elements in chain (I) have the same weight α1 = (1, 1, 2). Elements in chain (II) have the same weight α2 = (1, 1, −1). Elements in chain (III) have the same weight α3 = (1, −1, 0). (X1, X2, X3)((0, 1, 0) − (1, 0, 0)) = ((0, 1, 0) − (1, 0, 0), (1, 0, 0) − (0, 1, 0), (0, 1, 0) + (0, 0, 1) − (0, 0, 1) − (1, 0, 0)) = α3((0, 1, 0) − (1, 0, 0))

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 34 / 41

Gelfand-Tsetlin diagrams

A Gelfand-Tsetlin diagram is an array of integers of the form such that λ(i+1)

j

≥ λ(i)

j

≥ λ(i+1)

j+1

for every such triangle in the diagram.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 35 / 41

Gelfand-Tsetlin diagrams

Example (Gelfand-Tsetlin diagrams)

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 36 / 41

Remark

What is the relation between Gelfand-Tsetlin basis and Gelfand-Tsetlin diagrams? Is there any bejection among branching graph/ Gelfand-Tsetlin diargam/ Young tableaux? What can we say about the branching graph of symmetric group sequence? If we fix a representation λ at level n, and consider the interval from φ to λ, is it a distributive lattice (x ∧ (y ∨ z) = (x∧) ∨ (x ∧ z))? If the groups sequence {1} = G(0) ⊂ G(1) ⊂ G(2) ⊂ ... is a normal groups sequences. That is, G(i) is the normal subgroup of G(i + 1), what can we say about its branching graph?

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 37 / 41

Remark

This READING REPORT is based on the paper Murali K. Srinivasan, Symmetric chains, Gelfand-Tsetlin chains and the Terwilliger algebra of the binary Hamming scheme, Journal of Algebraic Combinatorics, 34(2), 301-322. and the references listed at the end. I am grateful to Prof. Hajime Tanaka for several helpful discussions.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 38 / 41

References

• N. G. de Bruijin, C.A. v. E. Tengbergen, D.Kruyswijk. (1951)

On the set of divisors of a number Nieuw Arch. Wiskunde 23, 191 – 193.

• A. M. Vershik, A. Okounkov. (2005)

A new approach to the representation theory of the symmetric groups - II Journal of Mathematical Sciences (New York) 131, 5471 – 5494. Ceccherini-Silberstein T, Scarabotti F, Tolli F. (2010) Representation theory of the symmetric groups: the Okounkov-Vershik approach, character formulas, and partition algebras Cambridge University Press Vol. 121. Bannai E, Ito T. (1984) Algebraic combinatorics Menlo Park: Benjamin/Cummings Terwilliger P. (1992) The subconstituent algebra of an association scheme,(Part I) Journal of Algebraic Combinatorics 1(4): 363-388.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 39 / 41

References

Srinivasan M K. (2011) Symmetric chains, Gelfand-Tsetlin chains, and the Terwilliger algebra of the binary Hamming scheme Journal of Algebraic Combinatorics 34(2), 301-322. Go J T. (2002) The Terwilliger algebra of the hypercube European Journal of Combinatorics 23(4), 399-429. Levstein F, Maldonado C, Penazzi D. (2006) The Terwilliger algebra of a Hamming scheme H(d, q) European Journal of Combinatorics 27(1), 1-10. Greene C, Kleitman D J. (1976) Strong versions of Sperner’s theorem Journal of Combinatorial Theory, Series A 20(1), 80-88. Billey S, Guillemin V, Rassart E. (2004) A vector partition function for the multiplicities of slkC Journal of Algebra 278(1), 251-293.

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 40 / 41

Thank You

Yizhe Zhu (SJTU) SJB, Terwilliger algebra, GZ basis December 13, 2014 41 / 41