By Kevin W. J. Kadell

ISBN-10: 0821825526

ISBN-13: 9780821825525

Macdonald and Morris gave a sequence of continuous time period $q$-conjectures linked to root structures. Selberg evaluated a multivariable beta kind essential which performs a massive function within the thought of continuous time period identities linked to root structures. Aomoto lately gave an easy and chic facts of a generalization of Selberg's indispensable. Kadell prolonged this evidence to regard Askey's conjectured $q$-Selberg vital, which was once proved independently by means of Habsieger. This monograph makes use of a continuing time period formula of Aomoto's argument to regard the $q$-Macdonald-Morris conjecture for the foundation method $BC_n$. The $B_n$, $B_n^{\lor}$, and $D_n$ instances of the conjecture stick with from the theory for $BC_n$. the various info for $C_n$ and $C_n^{\lor}$ are given. This illustrates the fundamental steps required to use tools given right here to the conjecture whilst the diminished irreducible root method $R$ doesn't have miniscule weight.

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4) [ l ] ( * - s ) S y m ( M ) = 0. 5) = [1] (t - s) Sym(M) -Q[l]-(ts =-Q[l]-(*-«)Sym(M). 6) [l]t(l - 7)(1 I - Q - ) S y m ( M ) = Q[l]i(tS s)Sym(s,t). 7) = [1] s (1 - j) Sym(s,t) - Q [1] (t - s) Sym(s,t) [l)S1(t-s)Sym(s,t). 7). Using /,. 1). 9) [i]I(* + 0(i-J)(l-Qi)Sym(«,0 = i [ i ] i ( « + *)(i-7)(i-Q7)Sym(« > 0. 9). • The following lemma, which we call the ^-transportation theory for ^4n-i> explicitly expresses Lemma 10 in terms of qbcn(ayb,k;ti,... ,< n ). Lemma 11. Let 2 < v < n. Ifu{t\,...

KADELL 28 L e m m a 7. tii ^-Jk;t\,... -QJ~ I qbc[qbc n(a,b, n(a,b,k;i ,*„) (1-g*) ^ (ti+tv) , , , ,, , . , , v "" M — ^ X > /y __ jbf \ r C n l f l > 0 ' *5*2>- •• , * v - l i * l , t v , . . *« + 1 ) . , . , , + 7; -T- > , 77-; - f 7 qbcn(a,b,k;t2l...

Proof. We proceed by induction on m. 1). 7) holds for a given m with m > 0 for all r > 1. a+2»+,+(„_m_2)* ( 1 _^ g f l ) qBCnm>r(a, b, k) ? B C n mr_l(a)6) fc). jBCn,m,r+1(a, 6, *)) *, *) o\ (,SC,n,m,r_i(a)6,fc)-9a+("-m-r-1^\BC*n,m,r(a,6,*)) q2b+l + (r-2)k ^ _ ? (r-l)tj x ( , B C n , m , r . 2 ( a , 6 , Jb) - q'+(n-m-r+i)k _,«+a»+l+(n-m-2)fc ( 1 _ ? * ) flC tBCn,m,r-i(a, (a>6tjfc)> r > L b, k)) 52 KEVIN W. J. 10) Q_ n _ a+2M-l+(2n-m-r-2)*\ (qBCntm,r(a,b,k)-qa+^-m-r-1)k 6> *)) ( j _ g(r-l)fc) x (,BC„,m,r_2(a)6,ib)-ga+(n-m-'-,-1)\BCn,m,r_1(a)6)Jb))) r > l .

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