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Determinental formulae for complete symmetric functions

David M Bressoud, Shi-Yuan Wei
1992 Journal of combinatorial theory. Series A  
Equations expressing each S,,j. as a determinant involving complete symmetric functions are well known [4] .  ...  We let h, = h,(xl, . . . . x,) denote the complete symmetric function of degree m in the variables xi, x2, . . . . x, (the sum of all distinct monomials of degree m in these variables).  ... 
doi:10.1016/0097-3165(92)90008-i fatcat:e4cdsunhsvaslconlfk2guqj2q

Ensemble Kernel Methods, Implicit Regularization and Determinantal Point Processes [article]

Joachim Schreurs, Michaël Fanuel, Johan A. K. Suykens
2020 arXiv   pre-print
Our first empirical results indicate that ensemble of ridgeless regressors can be interesting to use for datasets including redundant information.  ...  Connections between RLS, DPPs and Christoffel functions were explored in [FSS19] . Note that guarantees for DPP sampling for coresets have been derived in [TBA19] .  ...  Analogous result for kDPP sampling The elementary symmetric polynomial e k (K) is proportional to the (n − k)-th coefficients of the characteristic polynomial det(tI − K) = n k=0 (−1) k e k (K)t n−k .  ... 
arXiv:2006.13701v3 fatcat:mq7gejrwdvf67m2mwng5j6zjoa

Correlation Functions for β=1 Ensembles of Matrices of Odd Size

Christopher D. Sinclair
2009 Journal of statistical physics  
Using the method of Tracy and Widom we rederive the correlation functions for \beta=1 Hermitian and real asymmetric ensembles of N x N matrices with N odd.  ...  Introduction The starting point for many results concerning the spectral theory of random matrices is the derivation of a determinental or Pfaffian form for the correlation functions of the eigenvalues  ...  When β = 2, Fubini's Theorem together with elementary row and column operations on the Vandermonde determinant in the integrand lead to the determinental formula R n (λ) = det [K N (λ j , λ k )] n j,k=  ... 
doi:10.1007/s10955-009-9771-8 fatcat:u6jri7tb6ratnm36gygs23szai

Schur-finiteness in λ-rings [article]

C. Mazza, C. Weibel
2011 arXiv   pre-print
The authors would like to thank Anders Buch, Alessio Del Padrone and Christophe Soulé for valuable discussions.  ...  Symmetric functions We devote this section to a quick study of the ring Λ of symmetric functions, and especially the Schur polynomials s π , referring the reader to [Macd] for more information.  ...  Let A be an idempotent-complete exact category which is a QTC for ⊗. For any object A in A, the symmetric group Σ n (and hence the group ring Q[Σ n ]) acts on the n-fold tensor product A ⊗n .  ... 
arXiv:1011.1444v2 fatcat:znwcdovku5dc7et4vcnoqnunoy

Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture

Persi Diaconis
2003 Bulletin of the American Mathematical Society  
These same patterns appear in telephone encryption, the zeros of Riemann's zeta function, a variety of physics problems, and in the study of Toeplitz operators.  ...  Change of basis formulae between these two sets of symmetric functions give some information.  ...  Of course, the coefficients are just the elementary symmetric functions in the eigenvalues, and the traces are the power sum symmetric functions.  ... 
doi:10.1090/s0273-0979-03-00975-3 fatcat:fssgipqlcrcojdsgrcu7cjlrcq

Rational solutions of Painleve systems [article]

David Gomez-Ullate, Yves Grandati, Robert Milson
2020 arXiv   pre-print
The upshot of this analysis is a an explicit determinental representation for rational solutions in terms of classical orthogonal polynomials.  ...  Although the solutions of Painlevé equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions, there do exist rational solutions for specialized values  ...  The classical Jacobi-Trudi identity is a determinental representation of the Schur polynomials in terms of complete symmetric polynomials. Proposition 4.  ... 
arXiv:2009.11668v1 fatcat:y5ptpi5kfbfa7brnlrdedgwhwi

COMBINATORICS OF THE MODULAR GROUP II THE KONTSEVICH INTEGRALS

C. ITZYKSON, J.-B. ZUBER
1992 International Journal of Modern Physics A  
We study algebraic aspects of Kontsevich integrals as generating functions for intersection theory over moduli space and review the derivation of Virasoro and KdV constraints. 1.  ...  The main theorem 2.2 Expansion of Z on characters and Schur functions 2.3 Proof of the first part of the Theorem 3. From Grassmannians to KdV 4.  ...  Bauer for his assistance in algebraic calculations as well as in the elaboration of Lemma 5 and P. Ginsparg for a critical reading of the manuscript.  ... 
doi:10.1142/s0217751x92002581 fatcat:45mvst3we5af3iaexnyu3kwbhe

Wishart and anti-Wishart random matrices

Romuald A Janik, Maciej A Nowak
2003 Journal of Physics A: Mathematical and General  
We provide a compact exact representation for the distribution of the matrix elements of the Wishart-type random matrices A^† A, for any finite number of rows and columns of A, without any large N approximations  ...  This representation is of interest for a procedure of reconstructing the redundant information hidden in Wishart matrices, with potential applications to numerous models based on biological, social and  ...  Joint eigenvalue distributions In this section, for completeness, we give the results for the joint eigenvalue distributions for Wishart and Anti-Wishart random matrices.  ... 
doi:10.1088/0305-4470/36/12/343 fatcat:nplkgg6k3bcxdcsuxlvonathoi

Correlations for the Novak process [article]

Eric Nordenstam, Benjamin Young
2012 arXiv   pre-print
We study random lozenge tilings of a certain shape in the plane called the Novak half-hexagon, and compute the correlation functions for this process.  ...  The most difficult step in the present paper is to compute the inverse of the matrix whose (i,j) entry is the binomial coefficient C(A, B_j - i) for indeterminate variables A and B_1, ..., B_n.  ...  It is not at all clear how to algebraically relate (22) with the formula in [Joh05b, Theorem 3.1], since the latter is a sum involving products of Hahn polynomials.  ... 
arXiv:1201.4138v1 fatcat:wz75ce2uungc3hwzccfci2vfwq

Witt vectors. Part 1 [article]

Michiel Hazewinkel
2008 arXiv   pre-print
(t) (9.59) From (9.59) one readily obtains a formula for the power sum symmetric functions in terms of the complete symmetric functions, i.e. a universal formula for the ghost components of an element  ...  These determinental formulas are exactly the same as those linking power sums and elementary symmetric functions in symmetric function theory ([281] , p. 28), which is as must be because the defining  ...  Every symmetric power series can be uniquely written as a power series in the complete symmetric functions.  ... 
arXiv:0804.3888v1 fatcat:m2awczhf35bkpen2jpzvrwcgwa

Integral Geometry on Grassmann Manifolds and Calculus of Invariant Differential Operators

Tomoyuki Kakehi
1999 Journal of Functional Analysis  
In particular, we will give eigenvalue formulas and radial part formulas for invariant differential operators. Academic Press  ...  In this paper, we mainly deal with two problems in integral geometry, the range characterizations and construction of inversion formulas for Radon transforms on higher rank Grassmann manifolds.  ...  We remark that the above inversion formula is the generalization of the Helgason's inversion formula for the Radon transform on the real projective space P n&1 R. (See Helgason [H4].)  ... 
doi:10.1006/jfan.1999.3459 fatcat:pjkezq3blbdinezw5kfpmhq6om

General solutions for tunneling of scalar fields with quartic potentials

Fred C. Adams
1993 Physical Review D, Particles and fields  
We find the solutions numerically and use polynomial fitting formulae to obtain expressions for the Euclidean action.  ...  For the theory of a single scalar field φ with a quartic potential V(φ), we find semi-analytic expressions for the Euclidean action in both four and three dimensions.  ...  Watkins for useful discussions. This work was supported by NASA Grant No. NAGW-2802 and by funds from the Physics Department at the University of Michigan.  ... 
doi:10.1103/physrevd.48.2800 pmid:10016526 fatcat:mwxqikipavbdhchlzb6hhtn4z4

Weighted permutation problems and Laguerre polynomials

Richard Askey, Mourad E.H Ismail, Tom Koornwinder
1978 Journal of combinatorial theory. Series A  
The connection between integrals of products of Laguerre polynomials, power series coefficients of certain rational functions of several variables, and certain numbers of weighted permutation problems  ...  For a: > --I, let A(n, ,..., nk ; a) = (-l)nl+".+n* s m L",,(x) *.* L",,(x) x"e-u2 -2a, -... -(k -1) u*)a+r ' where CT~ is the jth elementary symmetric function in r1 ,..., rk .  ...  Formula (2.7) can be used to decrease any of the Ai's.  ... 
doi:10.1016/0097-3165(78)90020-1 fatcat:zkwilzpxuvbjdf22qej6z7dvqm

Correlations for the Novak process

Eric Nordenstam, Benjamin Young
2012 Discrete Mathematics & Theoretical Computer Science  
International audience We study random lozenge tilings of a certain shape in the plane called the Novak half-hexagon, and compute the correlation functions for this process.  ...  The most difficult step in the present paper is to compute the inverse of the matrix whose (i,j)-entry is the binomial coefficient $C(A, B_j-i)$ for indeterminate variables $A$ and $B_1, \dots , B_n.$  ...  It is not at all clear how to algebraically relate (22) with the formula in [Joh05b, Theorem 3.1], since the latter is a sum involving products of Hahn polynomials.  ... 
doi:10.46298/dmtcs.3070 fatcat:zpldz75w5vatnhsvtuthb7et3i

Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories [article]

Lara B. Anderson, James Gray, Andre Lukas, Burt Ovrut
2013 arXiv   pre-print
Thus, we have exchanged the problem of computing bundle holomorphy as a function of complex structure for that of computing cohomology groups of line bundles as a function of those variables.  ...  We recall that the dimension of a Young tableau may be easily computed from the hook-length formula (see [63] , for example).  ... 
arXiv:1304.2704v1 fatcat:m6f6optj2ja4zgu7o55h6pip7a
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