A cellular basis of the q-Brauer algebra Nguyen Tien Dung Vinh - PowerPoint PPT Presentation

Vinh University A cellular basis of the q-Brauer algebra Nguyen Tien Dung Vinh University 09 Sep, 2014 Dung Nguyen Tien

Vinh University 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 eg − 1 2 e = q − 1 e ; ( E 3 ) e (2) = g 2 g 3 g − 1 1 g − 1 2 e (2) = e (2) g 2 g 3 g − 1 1 g − 1 2 , where e (2) = e ( g 2 g 3 g − 1 1 g − 1 2 ) e . The elements e ( k ) in Br n ( r , q ) are defined inductively by e (1) = e and by e ( k +1) = eg + 2 , 2 k +1 g − 1 , 2 k e ( k ) . Dung Nguyen Tien

Vinh University 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 q − q − 1 e ; ( E 2 ) eg i = g i e for i > 2 , eg 1 = g 1 e = q 2 e , eg 2 e = rqe and eg − 1 2 e = ( rq ) − 1 e ; ( E 3 ) g 2 g 3 g − 1 1 g − 1 2 e (2) = e (2) g 2 g 3 g − 1 1 g − 1 2 . Dung Nguyen Tien

Vinh University 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 2 k +1 , n = F { s 2 k +1 , s 2 k +2 , · · · , s n − 1 } (the symmetric group) H 2 k +1 , n = F { g s , s ∈ S 2 k +1 , n } (the Hecke algebra) S λ : The Young subgroup of S 2 k +1 , n Std ( λ ): The set of all standard λ - tableaux Λ n := { ( k , λ ) | λ is a partition of n − 2 k } 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 } c µ = � m µ = e ( k ) c µ = c µ e ( k ) ; σ ∈ S µ g σ λ ˇ Br n := � � � ( s , u ) , ( t , v ) ∈ I n ( l , µ ) x µ � ( s , u )( t , v ) := g ∗ u g ∗ d ( s ) m µ g d ( t ) g v � µ ✄ λ for ( l , µ ) , ( k , λ ) ∈ Λ n � Dung Nguyen Tien

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Vinh University Example The Murphy basis of H 3 , 5 : { c st = g ∗ d ( s ) c λ g d ( t ) } 5 , we have c (1 3 ) 3 With t = , s = , p = 3 4 5 , q = = 1, 3 4 3 5 4 qq 5 4 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 = q 2 − 1 c ts + 1 q 2 c pp − 1 q 2 c tt − 1 q 2 c st − 1 q 2 c ss + c qq q 2 Dung Nguyen Tien

Vinh University Example The Murphy basis of H 3 , 5 : { c st = g ∗ d ( s ) c λ g d ( t ) } 5 , we have c (1 3 ) 3 With t = , s = , p = 3 4 5 , q = = 1, 3 4 3 5 4 qq 5 4 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 = q 2 − 1 c ts + 1 q 2 c pp − 1 q 2 c tt − 1 q 2 c st − 1 q 2 c ss + c qq q 2 Dung Nguyen Tien

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

Vinh University 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 ) Dung Nguyen Tien

Vinh University 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 3 q 5 ( r 2 − q 2 ) 2 ( q 4 r 2 − 1) � = 0 r 3 ( q 2 − 1) 3 3 Br 2 ( r , q ) is semisimple < = > e ( q ) > 2. 4 Br 3 ( r , q ) is semisimple < = > e ( q ) > 3 and 3 q ( r − q ) 2 ( q 2 r − 1) � = 0 ( q − 1) 3 5 Br 2 ( N ) is semisimple < = > e ( q ) > 2. 6 Br 3 ( N ) is semisimple < = > e ( q ) > 3 and 3 q 4 ( q N − q [ N ])([ N ] + q N +1 + q N +3 ) � = 0 Dung Nguyen Tien

Vinh University 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 ) Br 3 ( r 2 , q 2 ) is semisimple B 3 is not 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 ( r , q ) ∈ ( { ¯ 1 , ¯ 2 , ¯ 3 , ¯ 4 } × { ¯ 2 , ¯ 3 } ) ∪ ( { ¯ The BMW-algebra B 2 2 , The q -Brauer algebra Br 2 ( r 2 , q 2 ) ( r , q ) ∈ { ¯ 2 , ¯ 3 } × { ¯ 2 , ¯ 3 } ( r , q ) ∈ { ¯ 2 , ¯ 3 , ¯ 4 } × { ¯ The q -Brauer algebra Br 2 ( r , q ) 4 } Dung Nguyen Tien

Vinh University Thank for your attention Dung Nguyen Tien