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Macdonald's Symmetric Polynomials as Zonal Spherical Functions on Some Quantum Homogeneous Spaces
"... this paper we introduce some quantum analogues of the homogeneous spaces GL(n)=SO(n) and GL(2n)=Sp(2n) in the framework of quantum general linear groups. On these "quantum homogeneous spaces", we investigate the zonal spherical functions associated with finite dimensional representations. ..."
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Cited by 71 (2 self)
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this paper we introduce some quantum analogues of the homogeneous spaces GL(n)=SO(n) and GL(2n)=Sp(2n) in the framework of quantum general linear groups. On these "quantum homogeneous spaces", we investigate the zonal spherical functions associated with finite dimensional representations. As a result, we will see that the zonal spherical functions in question are represented by Macdonald's symmetric polynomials P = P (x 1 ; \Delta \Delta \Delta ; xn ; q; t) (of type An\Gamma1 ) with t = q or t = q ([M2]). This result can be regarded as a generalization of Koornwinder's realization of the continuous qLegendre polynomials by the quantum group SU q (2) ([K1]). Our quantum analogue of GL(n)=SO(n) is essentially the same as the one discussed by UenoTakebayashi [UT]. As to the quantum analogue of GL(3)=SO(3), it is already known by [UT] that Macdonald's symmetric polynomials arise as zonal spherical functions. Our result contains the affirmative answer to their conjecture for the case where n ? 3. Main results of this paper are announced in [N3]. Throughout this paper, we will denote by G the general linear group GL(N) and by g its Lie algebra gl(N ). The qdeformation of the coordinate ring A(G) of G = GL(N) and the universal enveloping algebra U(g) of g = gl(N) will be denoted by A q (G) and by U q (g), respectively. We consider the following closed subgroup K of G for "quantization": = Jn g (N = 2n); e 2k\Gamma1;2k \Gamma e 2k;2k\Gamma1 . The corresponding Lie subalgebra of g will be denoted by k. After the preliminaries on the quantum general linear groups (A q (G) and U q (g)), we introduce in Section 2 some coideals k q of U q (g), corresponding to the Lie subalgebras k = so(n) ae gl(n) and k = sp(2n) ae gl(2n). The construction of k q is carried out i...
Brownian Motion in a Weyl Chamber, NonColliding Particles, and Random Matrices
, 1997
"... . Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An\Gamma1 , the symmetric group. For any starting posit ..."
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. Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An\Gamma1 , the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time t without having collided by time t. We show that the probability that there will be no collision up to time t is asymptotic to a constant multiple of t \Gamman(n\Gamma1)=4 as t goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group Bn gives a model of n independent particles with a wall at x = 0. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point ...
A New Class Of Symmetric Functions
 Publ. I.R.M.A. Strasbourg, 372/S20, Actes 20 S'eminaire Lotharingien
, 1988
"... Contents. 1. Introduction 2. The symmetric functions P (q; t) 3. Duality 4. Skew P and Q functions 5. Explicit formulas 6. The Kostka matrix 7. Another scalar product 8. Conclusion 9. Appendix 1. Introduction I will begin by reviewing briefly some aspects of the theory of symmetric functions. Thi ..."
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Cited by 71 (1 self)
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Contents. 1. Introduction 2. The symmetric functions P (q; t) 3. Duality 4. Skew P and Q functions 5. Explicit formulas 6. The Kostka matrix 7. Another scalar product 8. Conclusion 9. Appendix 1. Introduction I will begin by reviewing briefly some aspects of the theory of symmetric functions. This will serve to fix notation and to provide some motivation for the subject of these lectures. Let x 1 , : : : , xn be independent indeterminates. The symmetric group Sn acts on the polynomial ring Z[x 1 ; : : : ; xn ] by permuting the x's, and we shall write n = Z[x 1 ; : : : ; xn ] Sn for the subring of symmetric polynomials in x<F8
Combinatorial Hopf algebras and generalized DehnSommerville relations
, 2003
"... A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ: H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the u ..."
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Cited by 69 (16 self)
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A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ: H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the ubiquity of quasisymmetric functions as generating functions in combinatorics. We illustrate this with several examples. We prove that every character decomposes uniquely as a product of an even character and an odd character. Correspondingly, every combinatorial Hopf algebra (H, ζ) possesses two canonical Hopf subalgebras on which the character ζ is even (respectively, odd). The odd subalgebra is defined by certain canonical relations which we call the generalized DehnSommerville relations. We show that, for H = QSym, the generalized DehnSommerville relations are the BayerBillera relations and the odd subalgebra is the peak Hopf algebra of Stembridge. We prove that QSym is the product (in the categorical sense) of its even and odd Hopf subalgebras. We also calculate the odd subalgebras of various related combinatorial Hopf algebras: the MalvenutoReutenauer Hopf algebra of permutations, the
ALGEBRAIC ASPECTS OF INCREASING SUBSEQUENCES
 DUKE MATHEMATICAL JOURNAL VOL. 109, NO. 1
, 2001
"... We present a number of results relating partial CauchyLittlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number ..."
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Cited by 68 (10 self)
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We present a number of results relating partial CauchyLittlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial CauchyLittlewood sums, as well as new proofs of old formulae; relations of these expressions to orthogonal polynomials on the unit circle; and explicit bases for invariant spaces of the classical groups, together with appropriate generalizations of the straightening algorithm.
Functional relations in solvable lattice models. I. Functional relations and representation theory
 Internat. J. Modern Phys. A
, 1994
"... Abstract. Reported are two applications of the functional relations (Tsystem) among a commuting family of rowtorow transfer matrices proposed in the previous paper Part I. For a general simple Lie algebra Xr, we determine the correlation lengths of the associated massive vertex models in the anti ..."
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Abstract. Reported are two applications of the functional relations (Tsystem) among a commuting family of rowtorow transfer matrices proposed in the previous paper Part I. For a general simple Lie algebra Xr, we determine the correlation lengths of the associated massive vertex models in the antiferroelectric regime and central charges of the RSOS models in two critical regimes. The results reproduce known values or even generalize them, demonstrating the efficiency of the Tsystem.
Inhomogeneous lattice paths, generalized Kostka polynomials and A_n1 supernomials
"... Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works and by Kashiwara's theory of crystal bases we define ..."
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Cited by 65 (14 self)
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Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works and by Kashiwara's theory of crystal bases we define a statistic on paths yielding two novel classes of polynomials.
Computing the Singular Value Decomposition with High Relative Accuracy
 Linear Algebra Appl
, 1997
"... We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the a ..."
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Cited by 60 (12 self)
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We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, whichin general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as #nite element problems and quantum mechanics, it is the smallest singular values that havephysical meaning, and should be determined accurately by the data. Many recent papers have identi#ed special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite di#erent, motivating us to seek a co...
Linear functionals of eigenvalues of random matrices
 Trans. Amer. Math. Soc
, 2001
"... Abstract. Let Mn be a random n × n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of Mn to converge to a Gaussian limit as n →∞. By Fourier analysis, this result ..."
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Cited by 59 (6 self)
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Abstract. Let Mn be a random n × n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of Mn to converge to a Gaussian limit as n →∞. By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of Mn. For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of Mn are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices. 1.
A Bijection Between LittlewoodRichardson Tableaux And Rigged Configurations
 Selecta Math. (N.S
, 1999
"... We define a bijection from LittlewoodRichardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasiparticle expression for the generalized Kostka polynomials K#R (q) labeled by a partition # and a sequence of rectangles R. The ..."
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Cited by 58 (26 self)
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We define a bijection from LittlewoodRichardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasiparticle expression for the generalized Kostka polynomials K#R (q) labeled by a partition # and a sequence of rectangles R. The generalized Kostka polynomials are qanalogues of multiplicities of the irreducible GL(n, C)module V highest weight # in the tensor product V R L . 1.