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Session 18. Harmonic analysis, orthogonal expansions and Dunkl theory
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Generalized Brownian processes,
random matrices, \(\alpha\) determinant and positive
definite functions on Coxeter (permutation) groups
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Marek Bożejko, Institute of Mathematics, University of Wrocław, Poland
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In my talk we will consider the following subjects:
- Positive definite functions on Coxeter groups (W,S) connected with
(a) the length function \(|x| = \min\{k: x = s_{1} ... s_{k} , s_{j} \in S\),
and representation is minimal\(\}\) and we show that the function \(P(x,q) = q^{|x|}\) is positive
definite on W for q in interval \([-1,1]\),
(b) also for the "block" lenght function \(||x||\) = the number of different
generators in the representation of x as above, we prove that the function
\(Q(x,t) = t^{||x||}\) is positive definite on \(W\) for \(t\) in interval \([0,1]\).
- Applications: (a) The construction of \(q-CCR\) relations
\(A(f) A^*(g) - q A^*(g) A(f) = < f,g>\) for \(f,g\) in a Hilbert space,
for real \(q\) and complex \(|q|=1\), and connections with theta function of
Jacobi,
(b) Free infinitely divisibility of classical normal law \(N(0,1)\), the
distributions like \(1/\cosh\) and others classical Meixner laws.
- Markov random matrices and special positive definite functions on
infinite permutation group related to the number of "isolated" fixed points of
a permutation.
- Applications to positivity results for \(q\)-determinants and \(\alpha\)-determinants.
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References- S.Belinschi, M.Bożejko, F.Lehner, R.Speicher, The normal distribution is
free infinitely divisible, Adv.Math. 226(2011), 3677-3698.
- M.Bożejko, R.Speicher, Interpolations between bosonic and fermionic
relations given by generalized Brownian motion, Math.Z. 222(1996), 135-160.
- M.Bożejko, M.Guta, Functors of white noise associated to characters of
the infinite symmetric groups, Comm.Math.Phys. 229 (2002),209-227.
- M.Bożejko, T.Hasebe, On free infinitely divisibility for classical
Meixner distributions, Prob.Math.Stat. 33(2013), 363-375.
- M.Bożejko, T.Hirai, Gelfand-Raikov representations of Coxeter groups
associated to positive definite norm functions, Prob. Math.Stat.
34(2014).
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M.Bożejko, W.Bożejko, Generalized Gaussian processes and positivity of q-determinants and \(\alpha\)-determinants, Preprint, Wrocław 2014,12 pp.
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