Some Identities in the Twisted Group Algebra of Symmetric Groups

Transcription

1 Some Identities in the Twisted Group Algebra of Symmetric Groups Milena Sošić Department of Mathematics University of Rijeka Radmile Matejčić 2, Rijeka 51000, Croatia August 8, 2013

2 Let S n denote the symmetric group on n letters i.e S n is the set of all permutations of a set M = {1,2,...,n} equiped with a composition as the binary operation on S n (clearly, the permutations are regarded as bijections from M to itself); M. Sošić (University of Rijeka) Some identities August 8, / 21

3 Let S n denote the symmetric group on n letters i.e S n is the set of all permutations of a set M = {1,2,...,n} equiped with a composition as the binary operation on S n (clearly, the permutations are regarded as bijections from M to itself); X = {X ab 1 a,b n} be a set of n 2 commuting variables X ab ; M. Sošić (University of Rijeka) Some identities August 8, / 21

4 Let S n denote the symmetric group on n letters i.e S n is the set of all permutations of a set M = {1,2,...,n} equiped with a composition as the binary operation on S n (clearly, the permutations are regarded as bijections from M to itself); X = {X ab 1 a,b n} be a set of n 2 commuting variables X ab ; R n := C[X ab 1 a,b n] denote the polynomial ring i.e the commutative ring of all polynomials in n 2 variables X ab over the set C with 1 R n as a unit element of R n. C = the set of complex numbers. M. Sošić (University of Rijeka) Some identities August 8, / 21

5 Let S n denote the symmetric group on n letters i.e S n is the set of all permutations of a set M = {1,2,...,n} equiped with a composition as the binary operation on S n (clearly, the permutations are regarded as bijections from M to itself); X = {X ab 1 a,b n} be a set of n 2 commuting variables X ab ; R n := C[X ab 1 a,b n] denote the polynomial ring i.e the commutative ring of all polynomials in n 2 variables X ab over the set C with 1 R n as a unit element of R n. C = the set of complex numbers. S n acts on the set X as follows: g.x ab = X g(a)g(b). M. Sošić (University of Rijeka) Some identities August 8, / 21

6 Let S n denote the symmetric group on n letters i.e S n is the set of all permutations of a set M = {1,2,...,n} equiped with a composition as the binary operation on S n (clearly, the permutations are regarded as bijections from M to itself); X = {X ab 1 a,b n} be a set of n 2 commuting variables X ab ; R n := C[X ab 1 a,b n] denote the polynomial ring i.e the commutative ring of all polynomials in n 2 variables X ab over the set C with 1 R n as a unit element of R n. C = the set of complex numbers. S n acts on the set X as follows: g.x ab = X g(a)g(b). This action of S n on X induces the action of S n on R n given by g.p(...,x ab,...) = p(...,x g(a)g(b),...) for every g S n and any p R n. M. Sošić (University of Rijeka) Some identities August 8, / 21

7 Recall, the usual group algebra C[S n ] = { σ S n c σ σ c σ C of the symmetric group S n is a free vector space (generated with the set S n ), where the multiplication is given by ( σ S n c σ σ ) ( τ S n d τ τ ) = } σ,τ S n (c σ d τ )στ. Here we have used the simplified notation: στ = σ τ for the composition σ τ i.e the product of σ and τ in S n. M. Sošić (University of Rijeka) Some identities August 8, / 21

8 Now we define more general group algebra A(S n ) := R n C[S n ] a twisted group algebra of the symmetric group S n with coefficients in the polynomial ring R n. Here denotes the semidirect product. M. Sošić (University of Rijeka) Some identities August 8, / 21

9 Now we define more general group algebra A(S n ) := R n C[S n ] a twisted group algebra of the symmetric group S n with coefficients in the polynomial ring R n. Here denotes the semidirect product. The elements of the set A(S n ) are the linear combinations g i S n p i g i with p i R n. M. Sošić (University of Rijeka) Some identities August 8, / 21

10 Now we define more general group algebra A(S n ) := R n C[S n ] a twisted group algebra of the symmetric group S n with coefficients in the polynomial ring R n. Here denotes the semidirect product. The elements of the set A(S n ) are the linear combinations g i S n p i g i with p i R n. The multiplication in A(S n ) is given by (p 1 g 1 ) (p 2 g 2 ) := (p 1 (g 1.p 2 ))g 1 g 2 where g 1.p 2 is defined by: g.p(...,x ab,...) = p(...,x g(a)g(b),...) M. Sošić (University of Rijeka) Some identities August 8, / 21

11 Now we define more general group algebra A(S n ) := R n C[S n ] a twisted group algebra of the symmetric group S n with coefficients in the polynomial ring R n. Here denotes the semidirect product. The elements of the set A(S n ) are the linear combinations g i S n p i g i with p i R n. The multiplication in A(S n ) is given by (p 1 g 1 ) (p 2 g 2 ) := (p 1 (g 1.p 2 ))g 1 g 2 where g 1.p 2 is defined by: g.p(...,x ab,...) = p(...,x g(a)g(b),...) The algebra A(S n) is associative but not commutative. M. Sošić (University of Rijeka) Some identities August 8, / 21

12 Let I(g) = {(a,b) 1 a < b n, g(a) > g(b)} denote the set of inversions of g S n. Then to every g S n we associate a monomial in the ring R n defined by X g := =, X ab (a,b) I(g 1 ) a<b,g 1 (a)>g 1 (b) X ab which encodes all inversions of g 1 (and of g too). More generally, for any subset A {1,2,...,n} we will use the notation X A := X ab X ba = X {a,b}, (a,b) A A, a<b (a,b) A A, a<b because X {a,b} := X ab X ba. M. Sošić (University of Rijeka) Some identities August 8, / 21

13 Definition To each g S n we assign a unique element g A(S n ) defined by g := X g g. Theorem For every g 1,g 2 A(S n) we have g 1 g 2 = X(g 1,g 2 )(g 1 g 2 ), where the multiplication factor is given by X(g 1,g 2 ) =. (a,b) I(g 1 1 )\I((g 1g 2 ) 1 ) X {a,b} M. Sošić (University of Rijeka) Some identities August 8, / 21

14 Recall, X(g 1,g 2 ) = (a,b) I(g 1 1 )\I((g1g2) 1 ) X {a,b} Note that if l(g 1 g 2 ) = l(g 1 )+l(g 2 ). So we have X(g 1,g 2 ) = 1 g 1 g 2 = (g 1 g 2 ) where l(g) := CardI(g) is the lenght of g S n. M. Sošić (University of Rijeka) Some identities August 8, / 21

15 Recall, X(g 1,g 2 ) = (a,b) I(g 1 1 )\I((g1g2) 1 ) X {a,b} Note that if l(g 1 g 2 ) = l(g 1 )+l(g 2 ). So we have X(g 1,g 2 ) = 1 g 1 g 2 = (g 1 g 2 ) where l(g) := CardI(g) is the lenght of g S n. The factor X(g 1,g 2) takes care of the reduced number of inversions in the group product of g 1,g 2 S n. M. Sošić (University of Rijeka) Some identities August 8, / 21

16 Recall, g 1 g 2 = X(g 1,g 2 )(g 1 g 2 ) for every g 1,g 2 A(S n ) Example Let g 1 = 132, g 2 = 312 S 3. Then g 1 g 2 = 213, l(g 1 ) = 1, l(g 2 ) = 2, l(g 1 g 2 ) = 1. Note that g 1 1 = 132, g 1 2 = 231, so g 1 g 2 =(X 23 g 1 ) (X 13 X 23 g 2 ) = X 23 X 12 X 32 g 1 g 2 = X {2,3} X 12 g 1 g 2. On the other hand we have: (g 1 g 2 ) = X 12 g 1 g 2, since (g 1 g 2 ) 1 = 213. Thus we get g 1 g 2 = X {2,3}(g 1 g 2 ) and X(g 1,g 2 ) = X {2,3}. M. Sošić (University of Rijeka) Some identities August 8, / 21

17 Recall, g 1 g 2 = X(g 1,g 2 )(g 1 g 2 ) for every g 1,g 2 A(S n ) Example For g 1 = 132, g 2 = 231 we have g 1 g 2 = 321, l(g 1 ) = 1, l(g 2 ) = 2, l(g 1 g 2 ) = 3. Further g 1 1 = 132, g 1 2 = 312 and (g 1 g 2 ) 1 = 321, so we get: g 1 g 2 =(X 23 g 1 ) (X 12 X 13 g 2 ) = X 23 X 13 X 12 g 1 g 2, (g 1 g 2 ) = X 12 X 13 X 23 g 1 g 2. Thus we get g 1 g 2 = (g 1g 2 ) and X(g 1,g 2 ) = 1. M. Sošić (University of Rijeka) Some identities August 8, / 21

18 We denote by t a,b, 1 a b n the following cyclic permutation in S n : k 1 k a 1 or b+1 k n t a,b (k) := b k = a k 1 a+1 k b which maps b to b 1 to b 2 to a to b and fixes all 1 k a 1 and b+1 k n i.e ( ) 1... a 1 a a+1... b 1 b b+1... n t a,b = 1... a 1 b a... b 2 b 1 b+1... n M. Sošić (University of Rijeka) Some identities August 8, / 21

19 We denote by t a,b, 1 a b n the following cyclic permutation in S n : k 1 k a 1 or b+1 k n t a,b (k) := b k = a k 1 a+1 k b which maps b to b 1 to b 2 to a to b and fixes all 1 k a 1 and b+1 k n i.e ( ) 1... a 1 a a+1... b 1 b b+1... n t a,b = 1... a 1 b a... b 2 b 1 b+1... n t b,a := t 1 a,b i.e k t b,a (k) = k +1 a 1 k a 1 or b+1 k n a k b 1 k = b M. Sošić (University of Rijeka) Some identities August 8, / 21

20 Then the sets of inversions are given by I(t a,b ) = {(a,j) a+1 j b}, I(t b,a ) = {(i,b) a i b 1}. so the corresponding elements in A(S n ) have the form: t a,b = t a,b t b,a = X ib a i b 1 a+1 j b X aj t b,a. Observe: t a,a = id, where I(t a,a ) =. Denote: t a = t a,a+1 (= t a+1,a ), 1 a n 1 (the transposition of adjacent letters a and a+1). Then: t a = X aa+1t a, with I(t a) = {(a,a+1)}. M. Sošić (University of Rijeka) Some identities August 8, / 21

21 Recall, g 1 g 2 = X(g 1,g 2 )(g 1 g 2 ) for every g 1,g 2 A(S n ) Corollary For each 1 a n 1 we have (t a) 2 = X {a,a+1} id. Here we have used that t at a = id and X {a,a+1} = X aa+1 X a+1a. Corollary For each g S n, 1 a < b n we have g t b,a = a<j n, g(a)>g(j) X {g(j),g(a)} (gt b,a ). In the case g S j S n j, 1 j k n we have g t k,j = (gt k,j ). M. Sošić (University of Rijeka) Some identities August 8, / 21

22 Recall, g 1 g 2 = X(g 1,g 2 )(g 1 g 2 ) for every g 1,g 2 A(S n ) Corollary (Braid relations) We have (i) t a t a+1 t a = t a+1 t a t a+1 for each 1 a n 2, (ii) t a t b = t b t a for each 1 a,b n 1 with a b 2. Corollary (Commutation rules) We have (i) t m,k t p,k = (t k )2 t p,k+1 t m 1,k if 1 k m p n. (ii) Let w n (= nn 1 21) be the longest permutation in S n. Then for every g S n we have (gw n ) wn = wn (w n g) = g. X {a,b} a<b,g 1 (a)<g 1 (b) M. Sošić (University of Rijeka) Some identities August 8, / 21

23 Decompositions (from the left) of certain canonical elements in A(S n ) Observe first: for g S n g 1 S 1 S n 1 and 1 k 1 n such that g = g 1 t k1,1 Then g(k 1 ) =g 1 (t k1,1(k 1 )) = g 1 (1) = 1 implies k 1 = g 1 (1). M. Sošić (University of Rijeka) Some identities August 8, / 21

24 Decompositions (from the left) of certain canonical elements in A(S n ) Observe first: for g S n g 1 S 1 S n 1 and 1 k 1 n such that g = g 1 t k1,1 Then g(k 1 ) =g 1 (t k1,1(k 1 )) = g 1 (1) = 1 implies k 1 = g 1 (1). Subsequently, the permutation g 1 S 1 S n 1 can be represented uniquely as g 1 = g 2 t k2,2 with g 2 S 1 S 1 S n 2 and 2 k 2 n. Then g 1 (k 2 ) =g 2 (t k2,2(k 2 )) = g 2 (2) = 2 implies k 2 = g 1 1 (2). M. Sošić (University of Rijeka) Some identities August 8, / 21

25 Decompositions (from the left) of certain canonical elements in A(S n ) Observe first: for g S n g 1 S 1 S n 1 and 1 k 1 n such that g = g 1 t k1,1 Then g(k 1 ) =g 1 (t k1,1(k 1 )) = g 1 (1) = 1 implies k 1 = g 1 (1). Subsequently, the permutation g 1 S 1 S n 1 can be represented uniquely as g 1 = g 2 t k2,2 with g 2 S 1 S 1 S n 2 and 2 k 2 n. Then g 1 (k 2 ) =g 2 (t k2,2(k 2 )) = g 2 (2) = 2 implies k 2 = g 1 1 (2). By repeating the above procedure we get the following decomposition: g = t kn,n t kn 1,n 1 t kj,j t k2,2 t k1,1 = t kj,j. 1 j n M. Sošić (University of Rijeka) Some identities August 8, / 21

26 Recall, g = t kn,n t kn 1,n 1 t kj,j t k2,2 t k1,1. Example Let S 3 = {123,132,312,321,231,213} then in A(S 3 ) we have: 123 = t 3,3 t 2,2 t 1,1, 132 = t 3,3 t 3,2 t 1,1, 312 = t 3,3 t 3,2 t 2,1, 321 = t 3,3 t 3,2 t 3,1, 231 = t 3,3 t 2,2 t 3,1, 213 = t 3,3 t 2,2 t 2,1. M. Sošić (University of Rijeka) Some identities August 8, / 21

27 Recall, g = t kn,n t kn 1,n 1 t kj,j t k2,2 t k1,1. Example Let S 3 = {123,132,312,321,231,213} then in A(S 3 ) we have: 123 = t 3,3 t 2,2 t 1,1, 132 = t 3,3 t 3,2 t 1,1, 312 = t 3,3 t 3,2 t 2,1, 321 = t 3,3 t 3,2 t 3,1, 231 = t 3,3 t 2,2 t 3,1, 213 = t 3,3 t 2,2 t 2,1. Now, assume that α 3 = g S 3 g. Then we get the following product form: α3 = ( t ) 3,3 } =(id) ( t 3,2 +t 2,2) (t 3,1 +t 2,1 +t ) 1,1 {{}}{{}}{{} β1 β2 β3 of simpler elements β i, 1 i 3 of the algebra A(S 3). M. Sošić (University of Rijeka) Some identities August 8, / 21

28 The general situation in A(S n ): Definition For every 1 k n we define β n k+1 := t n,k +t n 1,k + +t k+1,k +t k,k = k m n t m,k. Theorem Let Then αn = g. g S n αn = β1 β2 βn = 1 k n 1 β n k+1. M. Sošić (University of Rijeka) Some identities August 8, / 21

29 In what follows we are going to introduce some new elements in the algebra A(S n ) by which we will reduce β n k+1, 1 k n. The motivation is to show that the element α n A(S n ) can be expressed in turn as products of yet simpler elements of the algebra A(S n ). Definition For every 1 k n 1 we define γ n k+1 := ( id t n,k) ( id t n 1,k ) ( id t k+1,k ), δn k+1 := ( id (t k) 2 t n,k+1) ( id (t k ) 2 t n 1,k+1) ( id (t k ) 2 t ) k+1,k+1 Recall, (t k) 2 = X {k,k+1} id (= X kk+1 X k+1k id) and t k+1,k+1 = id. Theorem For every 1 k n we have the following factorization β n k+1 = δ n k+1 (γ n k+1) 1. M. Sošić (University of Rijeka) Some identities August 8, / 21

30 Recall, α n = β 1 β 2 β n with β 1 = id βn k+1 = (γ ) δ n k+1 n k+1 1 ) ( γn k+1 (id t = n,k ) δn k+1 (id (t = k )2 t n,k+1 id t n 1,k ) ( id t k+1,k ( id (t k )2 t n 1,k+1 ) ) ( ) id (t k )2 t k+1,k+1 Example (The factorization of α 2 A(S 2 )) We have i.e α 2 = β 2 α2 = ( id (t 1) 2) (id t ) 1 2,1 M. Sošić (University of Rijeka) Some identities August 8, / 21

31 Recall, α n = β 1 β 2 β n with β 1 = id βn k+1 = (γ ) δ n k+1 n k+1 1 ) ( γn k+1 (id t = n,k ) δn k+1 (id (t = k )2 t n,k+1 id t n 1,k ) ( id t k+1,k ( id (t k )2 t n 1,k+1 ) ) ( ) id (t k )2 t k+1,k+1 Example (The factorization of α 3 A(S 3 )) We have where β2 = ( id (t 2) 2) (id t ) 1 3,2 α 3 = β 2 β 3 β3 = ( id (t 1) 2 t 3,2) ( id (t 1 ) 2) (id t 2,1 ) 1 (id t ) 1 3,1 M. Sošić (University of Rijeka) Some identities August 8, / 21

32 Recall, α n = β 1 β 2 β n with β 1 = id βn k+1 = (γ ) δ n k+1 n k+1 1 ) ( γn k+1 (id t = n,k ) δn k+1 (id (t = k )2 t n,k+1 id t n 1,k ) ( id t k+1,k ( id (t k )2 t n 1,k+1 ) ) ( ) id (t k )2 t k+1,k+1 Example (The factorization of α 4 A(S 4 )) We have where β 2 = ( id (t 3) 2) (id t 4,3 α 4 = β 2 β 3 β 4 ) 1 β 3 = ( id (t 2) 2 t 4,3) ( id (t 2 ) 2) (id t 3,2) 1 ( id t 4,2 ) 1, β4 = ( id (t 1) 2 t 4,2) ( id (t 1 ) 2 t 3,2) ( id (t 1 ) 2) (id t ) 1 2,1 (id t 1 ) 3,1) ( id t 1. 4,1 M. Sošić (University of Rijeka) Some identities August 8, / 21

33 Conclusion In order to replace the matrix factorizations (from the right) given in 1 by twisted algebra computation, we nead to consider similar factorizations (but from the left). Here we used factorizations from the left, because they are more suitable for computing constants in the algebra of noncommuting polynomials (this will be elaborated in a fortcoming paper). 1 S. Meljanac, D. Svrtan, Determinants and inversion of Gram matrices in Fock representation of q kl -canonical commutation relations and applications to hyperplane arrangements and quantum groups. Proof of an extension of Zagier s conjecture, arxiv:math-ph/ vl, 26 Apr S. Meljanac, D. Svrtan, Study of Gram matrices in Fock representation of multiparametric canonical commutation relations, extended Zagier s conjecture, hyperplane arrangements and quantum groups, Math. Commun., 1 (1996), M. Sošić (University of Rijeka) Some identities August 8, / 21