A cellular basis of the q-brauer algebra

Transcription

1 A cellular basis of the q-brauer algebra Nguyen Tien Dung 09 Sep, 2014

2 Wenzl (2012) Version that contains H n (q) Fix N Z \ {0}, let q and r be invertible elements. Moreover, assume that if q = 1 then r = q N. The q-brauer algebra Br n (r, q) is defined over the ring Z[q ±1, r ±1, ((r 1)/(q 1)) ±1 ] by generators g 1, g 2, g 3,..., g n 1 and e and relations (H) The elements g 1, g 2, g 3,..., g n 1 satisfy the relations of the Hecke algebra H n ; (E 1 ) e 2 = r 1 q 1 e; (E 2 ) eg i = g i e for i > 2, eg 1 = g 1 e = qe, eg 2 e = re and eg2 1 e = q 1 e; (E 3 ) e (2) = g 2 g 3 g1 1 g 2 1 e (2) = e (2) g 2 g 3 g1 1 g 2 1, where e (2) = e(g 2 g 3 g1 1 g 2 1 )e. The elements e (k) in Br n (r, q) are defined inductively by e (1) = e and by e (k+1) = eg + 2,2k+1 g 1,2k e (k).

3 Dung (2014) Version that contains H n (q 2 ) Let r and q be invertible elements over the ring Z[q ±1, r ±1, ( r r 1 q q 1 )±1 ]. Moreover, if q = 1 then assume that r = q N with N Z \ {0}. The q-brauer algebra Br n (r 2, q 2 ) over Z[q ±1, r ±1, ( r r 1 q q 1 )±1 ] is the algebra defined via generators g 1, g 2, g 3,..., g n 1 and e and relations (H) The elements g 1, g 2, g 3,..., g n 1 satisfy the relations of the Hecke algebra H n ; (E 1 ) e 2 = r r 1 e; q q 1 (E 2 ) eg i = g i e for i > 2, eg 1 = g 1 e = q 2 e, eg 2 e = rqe and eg2 1 e = (rq) 1 e; (E 3 ) g 2 g 3 g1 1 g 2 1 e (2) = e (2) g 2 g 3 g1 1 g 2 1.

4 Notations in Theorem 1 k an integer, 0 k [n/2] B k,n = {u B k l(d) = l(u) with d = e (k) u D k,n } S 2k+1,n = F{s 2k+1, s 2k+2,, s n 1 } (the symmetric group) H 2k+1,n = F{g s, s S 2k+1,n } (the Hecke algebra) S λ : The Young subgroup of S 2k+1,n Std(λ): The set of all standard λ- tableaux Λ n := {(k, λ) λ is a partition of n 2k} λ µ : if µ > λ or µ = λ and m i=1 λ i m i=1 µ i I n (k, λ) := {(s, u) : s Std(λ) and u B k,n } m µ = e (k) c µ = c µ e (k) ; c µ = σ S µ g σ ˇBr λ { n := } x µ (s,u)(t,v) := g u gd(s) m (s, u), (t, v) I µg d(t) g n (l, µ) v µ λ for (l, µ), (k, λ) Λ n

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6 Example The Murphy basis of H 3,5 : {c st = g d(s) c λg d(t) } With t = 3 4 5, s = 3 5 4, p = 3 4 5, q = 3 4 5, we have c (13 ) qq = 1, c tt = 1 + g 3, c ts = (1 + g 3 )g 4, c st = g 4 (1 + g 3 ), c ss = g 4 (1 + g 3 )g 4, c pp = 1 + g 3 + g 4 + g 3 g 4 + g 4 g 3 + g 4 g 3 g 4. The presentation of g π = g 3 g 4 in The Murphy basis of H 3,5 g π = g 3 g 4 = q2 1 q 2 c ts + 1 q 2 c pp 1 q 2 c tt 1 q 2 c st 1 q 2 c ss + c qq

7 Example The Murphy basis of H 3,5 : {c st = g d(s) c λg d(t) } With t = 3 4 5, s = 3 5 4, p = 3 4 5, q = 3 4 5, we have c (13 ) qq = 1, c tt = 1 + g 3, c ts = (1 + g 3 )g 4, c st = g 4 (1 + g 3 ), c ss = g 4 (1 + g 3 )g 4, c pp = 1 + g 3 + g 4 + g 3 g 4 + g 4 g 3 + g 4 g 3 g 4. The presentation of g π = g 3 g 4 in The Murphy basis of H 3,5 g π = g 3 g 4 = q2 1 q 2 c ts + 1 q 2 c pp 1 q 2 c tt 1 q 2 c st 1 q 2 c ss + c qq

8 Example The presentation of g π = g 3 g 4 in The Murphy basis of H 3,5 g π = g 3 g 4 = q2 1 q 2 c ts + 1 q 2 c pp 1 q 2 c tt 1 q 2 c st 1 q 2 c ss + c qq The presentation of g d = g u eg π g v in the cell basis of Br 5 (r 2, q 2 ) g d = g u eg π g v = q2 1 q 2 x (2,1) (t,u)(s,v) + 1 q 2 x (3) (p,u)(p,v) 1 q 2 x (2,1) (t,u)(t,v) 1 q 2 x (2,1) (s,u)(t,v) 1 q 2 x (2,1) (s,u)(s,v) + x (13 ) (q,u)(q,v), with x λ (s,u)(t,v) = g u ec st g v = g u g d(s) ec λg d(t) g v = g u g d(s) m λg d(t) g v

9 Notations in Theorem 2 F : A field of characteristic p rad(c(k, λ)) = {x C(k, λ) x, y λ = 0 for all y C(k, λ)} D(k, λ) = C(k, λ)/rad(c(k, λ)). d λµ = [C(k, λ) : D(l, µ)]: the composition multiplicity of D(l, µ) in C(k, λ)

10 A semisimplicity criteria of the q-brauer algebra for n = 2, 3 Let F be a field with char(f ) = p. Then, 1 Br 2 (r 2, q 2 ) is semisimple <=> e(q 2 ) > 2. 2 Br 3 (r 2, q 2 ) is semisimple <=> e(q 2 ) > 3 and 3q 5 (r 2 q 2 ) 2 (q 4 r 2 1) r 3 (q 2 1) Br 2 (r, q) is semisimple <=> e(q) > 2. 4 Br 3 (r, q) is semisimple <=> e(q) > 3 and 3q(r q) 2 (q 2 r 1) (q 1) Br 2 (N) is semisimple <=> e(q) > 2. 6 Br 3 (N) is semisimple <=> e(q) > 3 and 3q 4 (q N q[n])([n] + q N+1 + q N+3 ) 0

11 Ex1 Over field C, the q-brauer algebra and the BMW-algebra simultaneously depend on two parameters r and q. Calculation shows that C BMW-algebra q-brauer algebra (r, q 2 ) = (q 1, i) B 3 is not semisimple Br 3 (r 2, q 2 ) is semisimple (r, q) = (q 1, i i) B 3 is not semisimple Br 3 (r, q) is semisimple Ex2 Over field F with char(f) = 5. The total parameter values, such that the algebras are not semisimple, are summarized in the following table. The non-semisimple case F 5 F 5 The BMW-algebra B 2 (r, q) ({ 1, 2, 3, 4} { 2, 3}) ({ 2, The q-brauer algebra Br 2 (r 2, q 2 ) (r, q) { 2, 3} { 2, 3} The q-brauer algebra Br 2 (r, q) (r, q) { 2, 3, 4} { 4}

12 Thank for your attention