Scaffolds. A graph-based system for computations in Bose-Mesner algebras. William J. Martin

Transcription

1 A graph-based system for computations in Bose-esner algebras Department of athematical Sciences and Department of Computer Science Worcester Polytechnic Institute Algebraic Combinatorics Seminar Shanghai Jiao Tong University, Shanghai October 18, 2016

2 Introduction Exercises The Simplest All matrices here are square. Rows and columns are indexed by some finite set X. denotes the diagonal of matrix equals the trace of atrix N, as a second-order tensor, is represented as N The sum of all entries of N is N The ordinary matrix product of and N is encoded as a series reduction N Entrywise multiplication is a parallel reduction N.

3 Introduction Exercises Example X {1, 2, 3}, , N 0 1/3 1/3 1/3 0 1/3 0 1/3 0

4 Introduction Exercises Example X {1, 2, 3}, , N 0 1/3 1/3 1/3 0 1/3 0 1/3 0

5 Introduction Exercises Example X {1, 2, 3}, , N 0 1/3 1/3 1/3 0 1/3 0 1/3 0

6 Introduction Exercises Example X {1, 2, 3}, , N 0 1/3 1/3 1/3 0 1/3 0 1/3 0

7 Introduction Exercises Example X {1, 2, 3}, , N 0 1/3 1/3 1/3 0 1/3 0 1/3 0 N

8 Introduction Exercises Example X {1, 2, 3}, N /3 1/3 1/3 0 1/3 0 1/3 0, N 0 1/3 1/3 1/3 0 1/3 0 1/3 0

9 Introduction Exercises Example X {1, 2, 3}, N while /3 1/3 1/3 0 1/3 0 1/3 0 N, N 0 1/3 1/3 1/3 0 1/3 0 1/3 0

10 Introduction Exercises Example X {1, 2, 3}, N while /3 1/3 1/3 0 1/3 0 1/3 0 N 5/3, N 0 1/3 1/3 1/3 0 1/3 0 1/3 0

11 Introduction Exercises Example X {1, 2, 3}, N while N /3 1/3 1/3 0 1/3 0 1/3 0 N 5/3 But [ 1/3 2/3 2/3 ], N 0 1/3 1/3 1/3 0 1/3 0 1/3 0

12 Introduction Exercises Example, continued X {1, 2, 3}, , N 0 1/3 1/3 1/3 0 1/3 0 1/3 0 Products: N

13 Introduction Exercises Example, continued X {1, 2, 3}, , N 0 1/3 1/3 1/3 0 1/3 0 1/3 0 Products: N 1 1/3 1/3 /3 2 1/3 1 1/3 2 while N

14 Introduction Exercises Example, continued X {1, 2, 3}, , N 0 1/3 1/3 1/3 0 1/3 0 1/3 0 Products: N 1 1/3 1/3 /3 2 1/3 1 1/3 2 while N

15 Definition Association Schemes The Drumstick and the Wing Definition Let X be a finite set and let A be a subalgebra of at X (C). Suppose we are given A (di)graph G (V (G), E(G)) A subset R V (G) of red nodes, and a map from edges of G to matrices in A: w : E(G) A (edge weights) The scaffold S(G; R, w) is defined as the quantity S(G; R, w) ϕ:v (G) X e E(G) e(a,b) w(e) ϕ(a),ϕ(b) ϕ(r). This is an element of V R, so we say S(G; R, w) is a scaffold of order m R. r R

16 Definition Association Schemes The Drumstick and the Wing A New Piece of Terminology Why do I call them scaffolds? Others have referred to these as star-triangle diagrams. Terwilliger credits Arnold Neumaier for their introduction.

17 Definition Association Schemes The Drumstick and the Wing Definition, a bit more general... A at X (C) A (di)graph G (V (G), E(G)) A subset R V (G) of red nodes, and a map from edges of G to matrices in A: w : E(G) A (edge weights) a subset F V (G) of fixed nodes and a fixed function ψ : F X The scaffold S(G; R, w; F, ψ) is defined as the quantity S(G; R, w; F, ψ) ϕ:v (G) X ( a F )(ϕ(a)ψ(a)) e E(G) e(a,b) w(e) ϕ(a),ϕ(b) r R ϕ(r).

18 Definition Association Schemes The Drumstick and the Wing Basic Operations Deletion: J Contraction: I

19 Definition Association Schemes The Drumstick and the Wing Count Homomorphisms Example: If A is the adjacency matrix of a simple graph H, A A, and we take R and w(e) A for all e E(G), then S(G;, w) Hom(G, H) is the number of graph homomorphisms from G into H. For example, S(K 3 ;, A) A A A counts labelled triangles in H.

20 Definition Association Schemes The Drumstick and the Wing Partition Functions statistical mechanics graph theory E.g., Tutte polynomial is partition function of the Potts model Vaughan Jones: spin models (here X is a set of spins ) yield link invariants (w(e) W ± ) Triply Regular Association Scheme: A s T r,s,t i,j,k A i A r A k A t T r,s,t i,j,k A i A k A j A j

21 Definition Association Schemes The Drumstick and the Wing Algebra, Combinatorics and Knot Theory Donald Higman, Tatsuro Ito, and François Jaeger

22 Definition Association Schemes The Drumstick and the Wing Henceforth: A is a (Commutative) Bose-esner Algebra may be useful for other contexts, but for the rest of this talk, A is a Bose-esner algebra. Algebraically, a (commutative) association scheme is a vector space of matrices closed under ordinary multiplication, entrywise multiplication, and conjugate-transpose, and containing the identities, I and J, for both multiplications.

23 Definition Association Schemes The Drumstick and the Wing Commutative Association Schemes Combinatorially, an association scheme is an ordered pair (X, R) where X is a finite set and R {R 0,..., R d } is a partition of X into binary relations satisfying R contains the identity relation: R 0 {(a, a) a X } for each i, there is an i {0,..., d} such that R i R i {(b, a) (a, b) R i } there exist constants p k ij such that, whenever (a, b) R k, {c X (a, c) R i, (c, b) R j } p k ij p k ij pk ji

24 Definition Association Schemes The Drumstick and the Wing The Association Scheme of Symmetric Group S 3 X { (1), (12), (13), (23), (123), (132) } (1) (12) (1) (12) (123) (13) (123) (13) (132) (23) (132) (23) One Cayley graph for each conjugacy class C 0 {(1)}, C 1 {(12), (13), (23)}, C 2 {(123), (132)}

25 Definition Association Schemes The Drumstick and the Wing The Association Scheme of S 3 A , A , A

26 Definition Association Schemes The Drumstick and the Wing Two Bases for Bose-esner Algebra Schur idempotents {A 0, A 1,..., A d } (adjacency matrices) A i A j δ i,j A i d A i A j pij k A k k0 matrix idempotents {E 0, E 1,..., E d } (projections onto eigenspaces) E i E j δ i,j E i E i E j 1 d qij k E k X k0 where the structure constants q k ij are called Krein parameters and we know q k ij 0.

27 Definition Association Schemes The Drumstick and the Wing Definition, Specialized to Association Schemes Let (X, R) be an association scheme. Suppose we are given A (di)graph G (V (G), E(G)) A subset R V (G) of red nodes, and a map from edges of G to matrices in our Bose-esner algebra: w : E(G) A (edge weights) The scaffold S(G; R, w) is defined as the quantity S(G; R, w) ϕ:v (G) X e E(G) e(a,b) w(e) ϕ(a),ϕ(b) ϕ(r). r R

28 Definition Association Schemes The Drumstick and the Wing Fundamental Identities for Association Schemes 1) pij k 0 if and only if 0 A k A j A i 2) q k ij 0 if and only if E j 0 E i E k

29 Definition Association Schemes The Drumstick and the Wing The Drumstick Since E i E j 1 X d h0 qh ij E h we have (E i E j ) E k qk ij X E k. So E i E j E k E i E j E k qk E ij k X And this is zero whenever qij k 0. Our notation here implies that no edges of G are incident to the center node in the left diagram besides the three edges shown.

30 Definition Association Schemes The Drumstick and the Wing The Wing Dual to this is the following identity for intersection numbers: since ( d ) (A j A k ) A i pjk e A e A i pjk i A i, we have e0 A j A k A i A i p i jk

31 Isthmuses Dickie s Theorem: a j 0 implies a 1 0 Isthmus Lemma (Suzuki) Lemma Let (X, R) be a commutative association scheme. (I) If qjk e qe lm 0 for all e h, then E k El E m qh lm X E k E h E j E j (II) If qjk e qe lm 0 for all e h, then E k E l E k E h E l E j E m E j E m

32 Isthmuses Dickie s Theorem: a j 0 implies a 1 0 Suzuki Isthmus Lemma, Proof of (II): E k E l E k I E l E j E m E j E m d e0 E k E e E l E j E m E k E h E l E j E m

33 Isthmuses Dickie s Theorem: a j 0 implies a 1 0 Dual Isthmus Lemma Lemma Assume (X, R) is a commutative association scheme. (I) If p e hi pe jk 0 for all e l, then A i A k A i p l jk A l A h A j A h (II) If p e hi pe jk 0 for all e l, then A i A k A i A k A l A h A j A h A j

34 Isthmuses Dickie s Theorem: a j 0 implies a 1 0 P- and Q-Polynomial Schemes A (symmetric) association scheme (X, R) with relations {R i } is P-polynomial (or metric ) with respect to the ordering A 0,..., A d of its Schur idempotents if, with respect to this ordering, k > i + j p k ij 0 and k i + j p k ij 0

35 Isthmuses Dickie s Theorem: a j 0 implies a 1 0 P- and Q-Polynomial Schemes A (symmetric) association scheme (X, R) with relations {R i } is P-polynomial (or metric ) with respect to the ordering A 0,..., A d of its Schur idempotents if, with respect to this ordering, k > i + j p k ij 0 and k i + j p k ij 0 A (symmetric) association scheme (X, R) with primitive idempotents {E j } is Q-polynomial (or cometric ) with respect to the ordering E 0,..., E d if, with respect to this ordering, k > i + j q k ij 0 and k i + j q k ij 0

36 Isthmuses Dickie s Theorem: a j 0 implies a 1 0 Dickie s Theorem (1995, generalized by Suzuki 1998) Theorem (Dickie, Thm ) Suppose (X, R) is a cometric association scheme with Q-polynomial ordering E 0, E 1,..., E d. Write a j q j 1j. If 0 < j < d and a j 0, then a 1 0. Proof: 0 Ej Ej 0 X b j Ej since ( Ej+1)Ej qj 1,j+1 Ej X Ej Ej+1 Ej 0 X bj Ej+1 since q e j,1 q e 1,j+1 0 for any e j + 1 Ej Ej+1

37 Isthmuses Dickie s Theorem: a j 0 implies a 1 0 Proof, continued Ej So 0 Ej 1 Ej+1 since this is just a linear combination of zero terms. Ej+1 Ej This is the inner product of our tensor (which is zero) with E j 1 E 1, another tensor, order 3. E 1, Next, 0 Ej 1 Ej+1 Ej+1 since q e 1,j 1 q e 1,j+1 0 for any e j Ej

38 Isthmuses Dickie s Theorem: a j 0 implies a 1 0 Proof, continued Likewise, 0 Ej 1 Ej+1 since q e 1,j 1 q e 1,j+1 0 for any e j Ej+1 0 Ej 1 Ej+1 Ej+1 Using the entrywise product,. Now we expand E 1 E j 1 b j X E j + a j 1 X E j 1 + c j X E j and observe q1,j+1 e 0 for e < j. Since c j 0, we have 0 Ej Ej+1 Ej+1

39 Isthmuses Dickie s Theorem: a j 0 implies a 1 0 End of Sample Proof Ej 0 Ej+1 since q e 1,j q e 1,j+1 0 for any e j + 1 Ej Now we have a drumstick! Ej+1 and we know (E j E j+1 )E 1 q1 j,j+1 E 1 X with qj,j by the cometric property. So we have 0 So SU(E 1 E 1 E 1 ) 0 which tells us that q , or a 1 0.

40 Isthmuses Dickie s Theorem: a j 0 implies a 1 0 The End Thank You. Special thanks to Professor Eiichi Bannai, Professor Yaokun Wu, and to Yan Zhu.