June 29 2007
Hilbert series of Invariants
and
Constant Term Identities
Garsia -Musiker-Wallach-Xin-Zabrocki
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A basic result of invariant Theory
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When do we have we have a Basic set ?
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How do we compute the Hilbert series?
Such an integral often reduces to taking the
constant term of the integrand
(including the density of the measure)
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Πk = trace X k
Θk = trace U X k−1 V
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The Constant Term Problem
Proof
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Our Constant term can has an explicit evaluation!
How do we show this?
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A Remarkable Identity
Theorem
!
σ∈Sn
sign(σ)σ
"
#
1≤i<j≤n
=
n
!
1 − qi
i=1
!
σ∈Sn
sign(σ)σ
"
?
$
(xi − qxj ) =
#
1≤i<j≤n
1−q
$
(xi − xj ) =
!
1≤i<j≤n
n!
!
(xi − xj )
1≤i<j≤n
(xi − xj )
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The first reduction
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The next reduction
(proved by manipulations using the previous identity)
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Computing Constant terms by partial fraction
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Eliminating a Variable
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Here we will only use a particular case
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Let us now compute a Constant term
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To evaluate our constant term we also need
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Conclusion
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THE END