Transcript Noncommutative Biorthogonal Polynomials

Noncommutative
Biorthogonal
Polynomials
by Emily Sergel
Thanks to Prof Robert Wilson
Introduction to Orthogonal
Polynomials
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Suppose we have W(x) so that for any
polynomial f(x), ∫ f(x)W(x) dx ≥ 0
Then <f(x),g(x)> = ∫ f(x)g(x)W(x) dx defines
an inner product
A sequence of polynomials {pn(x)} is
orthogonal with respect to <·,·> if <pn(x),
pm(x)> = 0 whenever n≠m
The nth moment is ∫ xnW(x) dx
Introduction cont.
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Given a sequence {Si}, we can define an inner
product with ith moment Si and we can also create
a sequence of orthogonal polynomials in the
following way:
The Noncommutative Problem
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Suppose {Si} do not commute
We still want to create orthogonal
polynomials in this manner but we have a
problem with the determinant
Many properties of determinants rely
heavily on the commutativity of the matrix
entries
Solution!
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Much study has gone into developing an
appropriate analogue to determinants and
quasideterminants appear to be the correct
solution.
Quasideterminants preserve several
determinant properties:
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A matrix with 2 equal rows or columns has all
quasideterminants equal to 0
Cramer’s rule
and many more
Quasideterminants
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Each nxn matrix has n2 quasideterminants
The formula has some involved notation but the
following are good illustrations:
Constructing Noncommutative
Orthogonal Polynomials
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Just as we hoped, we can make functions
that are orthogonal by looking at a
quasideterminant of a particular matrix
The following sequence of polynomials
{πn(x)} is orthogonal with moments {Si}
Biorthogonal Polynomials
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Let Iab = ∫∫ xaybK(x,y) dα(x)dβ(y) for K
satisfying a few additional conditions
{Iab} is the set of bimoments with respect to
K(x,y)
Just as before, we can define an inner
product and create sets of polynomials with
nice properties
Biorthogonal Polynomials
cont.
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Define {pn(x)},{qn(y)} as follows:
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∫∫ pn(x)qm(y)K(x,y) dα(x)dβ(y) = 0 if n≠m
Recurrence relations
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In all three situations, the polynomials
generated obey recurrence relations.
For the first case, commutative orthogonal
polynomials, the relation is as follows:
pn+1(x) = (anx+bn)pn(x) + cnpn-1(x)
The recurrences in the other cases are more
complex, but in the same spirit.
Our goals
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To understand biorthogonal functions from
a more algebraic standpoint
To define something new –
noncommutative biorthogonal functions –
in a meaningful way
To prove that these new functions have
similar constructions and recurrence
relations.
References
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Bertola, M., M. Gekhtman, and J. Szmigielski. "Cauchy Biorthogonal
Polynomials." ArXiv.org. 16 April 2009. 11 June 2009 <arXiv:hepth/9407124v1>.
Gelfand, Israel, D. Krob, Alain Lascoux, B. Leclerc, V. S. Retakh, and J. Y.
Thibon. "Noncommutative Symmetric Functions." ArXiv.org. 20 July
1994. 11 June 2009 <arXiv:hep-th/9407124v1>.
"Orthogonal polynomial." Wikipedia, the free encyclopedia. 27 May
2009. 11 June 2009
<http://en.wikipedia.org/wiki/Orthogonal_polynomial>.
“Quasideterminant.” Wikipedia, the free encyclopedia. 13 May 2009.
11 June 2009 <http://en.wikipedia.org/wiki/Quasideterminant>.