A Proof of the Q-Macdonald-Morris Conjecture for Bcn by Kevin W. J. Kadell

By Kevin W. J. Kadell

Macdonald and Morris gave a chain of continuing time period $q$-conjectures linked to root platforms. Selberg evaluated a multivariable beta sort critical which performs a major function within the concept of continuous time period identities linked to root platforms. Aomoto lately gave an easy and stylish facts of a generalization of Selberg's crucial. Kadell prolonged this facts to regard Askey's conjectured $q$-Selberg indispensable, which used to be proved independently via 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 approach $BC_n$. The $B_n$, $B_n^{\lor}$, and $D_n$ instances of the conjecture keep on with from the theory for $BC_n$. a few of the info for $C_n$ and $C_n^{\lor}$ are given. This illustrates the fundamental steps required to use tools given the following to the conjecture whilst the decreased irreducible root approach $R$ doesn't have miniscule weight.

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37). 1) holds with r replaced by r — 1. 2) U * t w(ti,... '*») A n (a,6,fc;*i,... ,tn) (r r-3 3) =( . ,... ,t n ). 2). This is valid for r = 2 since the term involved is 0. We obtain tr r " 2 [1] —— J\U u(tu... 3) ,tn) qbcn(a,b,k;ti,... ,tn) «=i =(qk~q(r-l)k)[l] n * * " ^ ' - " >*n)*M«>M;*i,... ,tn) i= l r + g - 2 * " 1 - ^ - 2 ) * [1] J J tt- u(tu • • • ,*n) qbcn(a, 6,*; * i , . . , t n ) .

The following lemma explicitly expresses the geometry of the simple roots of Bn and Cn in terms of g 6c n (a,6, fe;ti,... ,* n )Lemma 6. g 6c n (a,6,fc;*2,-" ,tv, • • >*n) n * 2 . 58) X v-1 x II (^^W^iMjT-)*. 59) i=2 x 2 x i ,=2 *i I I (^U^kiUtM^-h, 2+i *J f l ( f ) a ( ^ ^ * i * l

We see that S y m ^ - i , tv) is symmetric in tv-\ and tv. 17) and s = *„_i, t = tVl Q = qk. 14), respectively. • The following lemma extends Lemma 10 from An-\ to Bn. Lemma 12. 18) T(j) be invariant under t <-+ l/t and have a Laurent expansion at t = 0. 19) [1] (1 + 1) (l - <)(1 - f ) T(<) = Q [1] (1 + < ) ( ! - 0(1 " j) T(0- Proof. 21) [l](l + i ) ( l - O T ( 0 = 0. 22) =[i](i + I)(i_0T(<)-Q[l]i(l = -Q[l}\(l + + I)(l-<)T(<) j)(l-t)T(t). 23) [i](i + i ) ( i - t ) ( i - 5 ) T ( o = g[i]<(i + y ) ( i - O T ( 0 = Q[l](l + t)(l-t)T(t).

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