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A Proof of the $q$-Macdonald-Morris Conjecture for $BC_n$

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Electronic ISBN: 978-1-4704-0093-4
Product Code: MEMO/108/516.E
List Price: $39.00 MAA Member Price:$35.10
AMS Member Price: $23.40 Click above image for expanded view A Proof of the$q$-Macdonald-Morris Conjecture for$BC_n$Available Formats:  Electronic ISBN: 978-1-4704-0093-4 Product Code: MEMO/108/516.E  List Price:$39.00 MAA Member Price: $35.10 AMS Member Price:$23.40
• Book Details

Memoirs of the American Mathematical Society
Volume: 1081994; 80 pp
MSC: 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 $q$-Macdonald-Morris 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 $q$-Macdonald-Morris conjecture for $B_n$, $B^\vee _n$, $C_n$, $C^\vee _n$ and $D_n$
• 13. Conclusion
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Volume: 1081994; 80 pp
MSC: 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 $q$-Macdonald-Morris 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 $q$-Macdonald-Morris conjecture for $B_n$, $B^\vee _n$, $C_n$, $C^\vee _n$ and $D_n$
• 13. Conclusion
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