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A Proof of the $q$-Macdonald-Morris Conjecture for $BC_n$
 
Front Cover for 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
Front Cover for A Proof of the $q$-Macdonald-Morris Conjecture for $BC_n$
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  • Front Cover for A Proof of the $q$-Macdonald-Morris Conjecture for $BC_n$
  • Back Cover for A Proof of the $q$-Macdonald-Morris Conjecture for $BC_n$
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.

    Readership

    Research 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 $q$-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.

Readership

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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