Electronic ISBN:  9781470400934 
Product Code:  MEMO/108/516.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 108; 1994; 80 ppMSC: Primary 17;
Macdonald and Morris gave a series of constant term \(q\)conjectures associated with root systems. Selberg evaluated a multivariable beta type integral which plays an important role in the theory of constant term identities associated with root systems. Aomoto recently gave a simple and elegant proof of a generalization of Selberg's integral. Kadell extended this proof to treat Askey's conjectured \(q\)Selberg integral, which was proved independently by Habsieger. This monograph uses a constant term formulation of Aomoto's argument to treat the \(q\)MacdonaldMorris conjecture for the root system \(BC_n\). The \(B_n\), \(B_n^{\lor }\), and \(D_n\) cases of the conjecture follow from the theorem for \(BC_n\). Some of the details for \(C_n\) and \(C_n^{\lor }\) are given. This illustrates the basic steps required to apply methods given here to the conjecture when the reduced irreducible root system \(R\) does not have miniscule weight.
ReadershipResearch mathematicians.

Table of Contents

Chapters

1. Introduction

2. Outline of the proof and summary

3. The simple roots and reflections of $B_n$ and $C_n$

4. The $q$engine of our $q$machine

5. Removing the denominators

6. The $q$transportation theory for $BC_n$

7. Evaluation of the constant terms $A$, $E$, $K$, $F$ and $Z$

8. $q$analogues of some functional equations

9. $q$transportation theory revisited

10. A proof of Theorem 4

11. The parameter $r$

12. The $q$MacdonaldMorris conjecture for $B_n$, $B^\vee _n$, $C_n$, $C^\vee _n$ and $D_n$

13. Conclusion


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Macdonald and Morris gave a series of constant term \(q\)conjectures associated with root systems. Selberg evaluated a multivariable beta type integral which plays an important role in the theory of constant term identities associated with root systems. Aomoto recently gave a simple and elegant proof of a generalization of Selberg's integral. Kadell extended this proof to treat Askey's conjectured \(q\)Selberg integral, which was proved independently by Habsieger. This monograph uses a constant term formulation of Aomoto's argument to treat the \(q\)MacdonaldMorris conjecture for the root system \(BC_n\). The \(B_n\), \(B_n^{\lor }\), and \(D_n\) cases of the conjecture follow from the theorem for \(BC_n\). Some of the details for \(C_n\) and \(C_n^{\lor }\) are given. This illustrates the basic steps required to apply methods given here to the conjecture when the reduced irreducible root system \(R\) does not have miniscule weight.
Research mathematicians.

Chapters

1. Introduction

2. Outline of the proof and summary

3. The simple roots and reflections of $B_n$ and $C_n$

4. The $q$engine of our $q$machine

5. Removing the denominators

6. The $q$transportation theory for $BC_n$

7. Evaluation of the constant terms $A$, $E$, $K$, $F$ and $Z$

8. $q$analogues of some functional equations

9. $q$transportation theory revisited

10. A proof of Theorem 4

11. The parameter $r$

12. The $q$MacdonaldMorris conjecture for $B_n$, $B^\vee _n$, $C_n$, $C^\vee _n$ and $D_n$

13. Conclusion