Partition algebra and party algebra

Transcription

1 Partition algebra and party algebra (Masashi KOSUDA) Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus Nishihara-cho Okinawa , Japan May 3, 2007 URL:

2 I. What is the party (partition) algebra? --- Centralizer of the unitary reflection group of type G(r,,k) in the tensor representation Unitary reflection group of type G(r,,k) < Permutation matrices of size k, ξ Diag(ζ,,, ) > (ζ; primitive r-th root of ) G(r,,k) ~ S k Diag(ζ i,ζ i2,,ζ ik ) G(,,k) ~ S k V C k : vector space on which G(r,,k) acts naturally G(r,,k) acts also on V Vby g(v v 2 v n ) gv gv 2 gv n (gg(r,,k)) P n,r (k) End G(r,,k) (V V): Party algebra In particular, if r, A n (k) : P,r (k) End Sk (V V): Partition algebra V.F.R.Jones, P.P.Martin (994) Remark) Since G(r,,k) k S, we have P n,r (k) n A(k)

3 Schur-Weyl duality GL k (C) O k (C),,k) G(r k S k n k n k n k n CS n n B(k) n,r (k) P n A(k) q-deform Iwahori BMW?? Hecke Find a basis for A n (k) (the partition algebra case) XA n (k) XEnd Sk (V V) { XEnd(V V) σ - Xσ σx for σ S k Put X( e m e m2 e mn ) Σ f,, fn X F m, mn e f e f n e f n F (f,, fn)[k] n [k] {, 2,, k} Then by σ - Xσ σx, we have σ(f),,σ( fn) X σ(m ),,σ( mn) f,, fn X m,, mn for σ S k

4 σ(f),,σ( fn) X σ(m ),,σ( a n+,, a 2n Xa,, a n mn) f,, fn X m,, mn for σ S k : a function on [k] 2n which takes the same values on the orbits for S k action. Example 3, X,3 (n2) 2, 2,3 X,2 X 3,2,2 X,3 2,3 3,2 X 2, X 3, Charcteristic functions on the orbits make a basis of A n (k) i.e. A n (k) < T ~ ~ is an equiv. rels. on [2n] #classes for ~ k > a n+,, a (T ~ ) 2n a,, a n { Interpretation of the basis by the diagram FM 2n 0 if ( ai aj i ~ j ) otherwise Σn,k {{T,, T s } s, 2,, k 2n, T j (φ) FM, T j FM, T j T j φ if ij }

5 Call wσn,k a seat-plan and T j w a table If k2n write Σn,k Σn Example : the diagram of a seat-plan of type Σn Σ5 {{f, m, m 2, m 4 }, {f 2 }, {f 3, f 4 }, {f 5, m 3 },{m 5 }} f f 2 f 3 f 4 f 5 T T2 T3 T4 T5 m m 2 m 3 m 4 m 5 k dim A n (k) #Σn,k ΣS(2n,i) ( B(2n) if k2n) i B(n) (), 2, (5), 5, (52), 203, Find a basis for P n,r (k) (the party algebra case) XP n,r (k) { XEnd(V V) σ - Xσ σx for σ S k ξx Xξ for ξ diag(ζ,,,) e m e m n e m n ξ( e m e m n e m n) ζ p ( ) p : total # of ''s in m, m 2,, m n

6 ξx Xξ ζ q X f,,fn m,,mn ζ p X f,,fn m,,mn q : total # of ''s in f, f 2,, f n q p ( mod r ) The same identity also holds for the total #s of 'i' s in f, f 2,, f n and m, m 2,, m n (i, 2,, k) i.e. the total # of 'i' s in f, f 2,, f n the total # of 'i' s in m, m 2,, m n ( mod r ) r (i, 2,, k) Σn,k {{T,, T s }Σn,k ; T j F T jm (mod r) (j, 2,, s)} In the following we assume that k >> Example : r > n T j F T j M each table has the same # of males ane females Example : r 2 T j F jm T (mod 2) the parity of #males and #females is the same on each table. each table has even number of the participants

7 Product in A n (k) and P n,r (k) f f 2 f 3 f 4 f 5 w w 2 f f 2 f 3 f 4 f 5 w k w 2 m m 2 m 3 m 4 m 5 m m 2 m 3 m 4 m 5 k Generators of A n (k) and P n,r (k) i i+ i i+ i i+ i+r- s i f [r] i e i v i v i+ v n v i+ v i v n s i : v v s i P P Σ Ejl Elj T u + T v j,l More generally, r Σn,k w Σ T v v w u v (v is coarser than w)

8 II. Irreducible reps. of the party (partition) algebra V V : G(r,,k)-P n,r (k) bimodule V V ~ S λ A λ : Schur-Weyl λ Decompose V Vas G(r,,k) bimodule. Decompse V n as G(,,k) ~ S k module V e, e 2,, e k > ~ < e + e e k > < e -e 2, e 2 -e 3,, e k- -e k > trivial rep. k reflection rep. ρ k Suppose V n- ~ ( m(µ)s µ ) µ Then we have V n ~ ( µ m(µ)s µ ) ( k ρ k ) Mackey's tensor product theorem (the last theorem which I learned in Shinoda's seminar) G H: finite groups M : G-module, N : H-module (M N) ~ M(N) G H G G H Putting G S k, H S k-, M S µ and N k-, H k-{

9 We obtaion S k S k S S k- S k- k- k S k- ) (S µ k-) ~ S µ ( Hence we have Examples ~ S k S µ Sk- S µ V ~ (S µ ) ~ S λ λ S k- S k S k S k- ~ k ρ k ~ V ( ) λ : obtained from µ by removing one box and then adding one box. ~ ( k ρ k ) V ~ VV~

10 Bratteli diagram for the partition algebra A 3 (6) 0-th -st 5 2-nd rd dim S λ dim A λ G(,,k) ~ S k -module A n (k) -module If k2n, then we have the same Bratteli diagram, the structure of A n (k) does not depend on the parameter k, replacing k with Q ( {0,, 2,, 2n-}), we can define A n (Q) ~ A n (k).

11 2. Decompse V n as G(r,,k) ~ S k Diag(ζ i,ζ i2,,ζ ik ) module Irrep. of G(r,,k) : λ (λ (), λ (2),, λ (r) ) λ λ () + λ (2) + + λ (r) k V < e, e 2,, e k > : irreducible and corresponds to ρ (,,φ,, φ) (if r > ) k-{ Decomposition of S µ Example n4, r3 V µ (, φ, ) ρ (,,φ) µ ρ (,, ) (,, ) Remark There occur no multiplicities in the decomposition S µ V (µ (µ (), µ (2),, µ (r) )) Remove one box from non-empty µ (i) and then add one box to µ (i+) (µ () if i r) so that it again becomes a Young diagram. (,φ,φ) (,φ,φ) (,φ,φ)

12 Bratteli digagram of P n,r (Q) Example: P 4,3 (5) φφ 0-th φ -st φ φ φ 2-nd φ φ φ φφ φφ φ φ φ φ φ φ φ φ φ Example: P 4,2 (5) φ 0-th -st φ φ 2-nd 3-rd φ φ φ φ

13 3. Remove multiplicities in the Bratteli diagram of A n (k) dim A -/2 (k) A 2-/2 (k) A 3-/2 (k) A 0 (k) A(k) 2 A(k) 3 A(k) B() B(2) B(3) B(4) B(5) B(6) A n-/2 (k) f f 2 f j f n- f n m m 2 m j m n- m n Σ n-/2 B(2n-) Bratteli diagram for A n- (k)a n-/2 (k)a n (k) {µ}; Irrep. of A n- (k) {λ}; Irrep. of A n (k) obtained from λ by removing one box {ν}; Irrep. of A n-/2 (k) f f 2 f j f n- f n m m 2 m j m n- m n and then adding one box.

14 Bratteli diagram for A n (6) : n 0, /2,, 3/2, 2, 5/2, 3 0-th -/2 Σ (dim S ν ) 2 ν -st 5 2-/2 2 4 Σ (dim S ν ) 2 5 ν 2-nd / Σ (dim S ν ) 2 52 ν 3-rd Σ (dim S ν ). (dim A ν ) (dim V) n- How is A n-/2 (k) mapped to End V n? Is A n-/2 (k) still centralizer of some algebra? If so, what does A n-/2 (k) centralize?

15 The dimension calculation sugests that A n-/2 (k) acts on V n- V n. The restrictions of Young diagrams sugests that A n-/2 (k) is a centralizer w.r.t. S k- action. A n-/2 (k) End Lk- (V n- ) Ram (2006) Here L k- ~ S k- {} acts diagonally on V n- ~ V n- e k < v v 2 v n- > e k V n Interpretation of Σn-/2,k in End V n Σn-/2,k w Σ T v vw (v is coarser than w) Here if v (n 5) {{f, m 3 }, {f 4, m }, {m 4 },{f 2, f 3, f 5, m 3, m 5 }}, then T v Σ X σ(2), k, σ(),σ(3), k σ(), k, k, σ(2), k σs k-

16 Now we (Naruse and I) are working on Description of the Murphy operators for A n (k), A n-/2 (k), P n,r (k), P n-/2,r (k),. Constructing the orthogonal representations of A n (k), A n-/2 (k), P n,r (k), P n-/2,r (k),. Cell representations, Character formulae (Naruse)

17 References [] W. Doran IV and D. Wales, The partition algebra revisited, J. Algebra 23 (2000), [2] J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 23 (996), 34. [3] V. F. R. Jones, The potts model and the symmetric group, Subfactors (Kyuzeso, 993) , World Sci. Publishing, River Edge, NJ, 994. [4] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (979), [5] T. Halverson and A. Ram, Partition algebras, European J. Combin. 26 (2005), [6] M. Kosuda, Characterization for the party algebras, Ryukyu Math. J. 3 (2000), [7] M. Kosuda, Irreducible representations of the party algebra, Osaka J. Math. 43 (2006), [8] M. Kosuda, The standard expression for the party algebra, Sūrikaisekikenkyūsho Kōkyūroku No. 497 (2006), [9] P. Martin, Representations of graph Temperley-Lieb algebras, Publ. Res. Inst. Math. Sci. 26 (990), [0] P. Martin, Temperley-Lieb algebras for non-planar statistical mechanics The partition algebra construction, J. Knot Theory Ramifications 3 (994), [] H. Naruse, Characters of the party algebras, manuscript for Workshop on Cellular and Diagram Algebras in Mathematics and Physics, Oxford, 2005.

18 [2] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (954), [3] K. Tanabe, On the centralizer algebra of the unitary reflection group G(m, p, n), Nagoya. Math. J. 48 (997), [4] C. C. Xi, Partition algebras are cellular, Compositio. Math. 9 (999),