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Elliptic determinant evaluations and the Macdonald identities for affine root systems
 Compositio Math
"... We obtain several determinant evaluations, related to affine root systems, which provide elliptic extensions of Weyl denominator formulas. Some of these are new, also in the polynomial special case, while others yield new proofs of the Macdonald identities for the seven infinite families of irreduci ..."
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We obtain several determinant evaluations, related to affine root systems, which provide elliptic extensions of Weyl denominator formulas. Some of these are new, also in the polynomial special case, while others yield new proofs of the Macdonald identities for the seven infinite families of irreducible reduced affine root systems. 1.
The Holonomic ansatz II: automatic discovery(!) and proof (!!) of the holonomic determinant evaluations
 Annals Comb
"... assume that readers have read [Z1]. ..."
An Application of Sobolev Orthogonal Polynomials to the Computation of a Special Hankel Determinant, Springer Optimization and Its Applications 42
, 2010
"... Abstract. Many Hankel determinants computations arising in combinatorial analysis, can be done by results from the theory of standard orthogonal polynomials. Here, we will emphasize a special sequence which requires including of discrete Sobolov orthogonality in order to find their closed form. 1 ..."
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Abstract. Many Hankel determinants computations arising in combinatorial analysis, can be done by results from the theory of standard orthogonal polynomials. Here, we will emphasize a special sequence which requires including of discrete Sobolov orthogonality in order to find their closed form. 1
Pfaffians and representations of the symmetric group
 in preparation. HJALMAR ROSENGREN
"... ooo ’ oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo o Pfaffians of matrices with entries z[i,j]/(xi + xj), or determinants of matrices with entries z[i,j]/(xi −xj), where the ant ..."
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Cited by 11 (3 self)
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ooo ’ oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo o Pfaffians of matrices with entries z[i,j]/(xi + xj), or determinants of matrices with entries z[i,j]/(xi −xj), where the antisymmetrical indeterminates z[i,j] satisfy the Plücker relations, can be identified with a trace in an irreducible representation of a product of two symmetric groups. Using Young’s orthogonal bases, one can write explicit expressions of such Pfaffians and determinants, and recover in particular the evaluation of Pfaffians which appeared in the recent literature. ooo ’ oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo ’ o ooo oo
Almost Product Evaluation of Hankel Determinants
"... An extensive literature exists describing various techniques for the evaluation of Hankel determinants. The prevailing methods such as Dodgson condensation, continued fraction expansion, LU decomposition, all produce product formulas when they are applicable. We mention the classic case of the Hanke ..."
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An extensive literature exists describing various techniques for the evaluation of Hankel determinants. The prevailing methods such as Dodgson condensation, continued fraction expansion, LU decomposition, all produce product formulas when they are applicable. We mention the classic case of the Hankel determinants with binomial entries �3k+2 � �3k � k and those with entries k; both of these classes of Hankel determinants have product form evaluations. The intermediate case, �3k+1 � k has not been evaluated. There is a good reason for this: these latter determinants do not have product form evaluations. In this paper we evaluate the Hankel determinant of �3k+1 � k. The evaluation is a sum of a small number of products, an almost product. The method actually provides more, and as applications, we present the salient points for the evaluation of a number of other Hankel determinants with polynomial entries, along with product and almost product form evaluations at special points. 1
PERIOD POLYNOMIALS AND EXPLICIT FORMULAS FOR HECKE OPERATORS ON Γ0(2)
, 2007
"... Abstract. Let Sw+2(Γ0(N)) be the vector space of cusp forms of weight w+2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2( ..."
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Abstract. Let Sw+2(Γ0(N)) be the vector space of cusp forms of weight w+2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2(Γ0(2)), and extend the EichlerShimuraManin isomorphism theorem to Γ0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on Sw+2(Γ0(2)). As an application of our main theorems, we will also give an affirmative answer to a speculation of Imamo¯glu and Kohnen on a basis of Sw+2(Γ0(2)). Let Γ be a congruence subgroup of SL2(Z). One of the most important problems in the theory of modular forms is to obtain explicit formulas for Hecke operators on cusp forms for Γ. When Γ is the full modular group SL2(Z), this was done in [8], where we gave explicit formulas in terms of Bernoulli numbers Bk and divisor
Generalizations of Cauchy’s determinant and Schur’s Pfaffian
, 2005
"... We present several identities of Cauchytype determinants and Schurtype Pfaffians involving generalized Vandermonde determinants, which generalize Cauchy’s determinant det(1/(xi + yj)) and Schur’s Pfaffian Pf ((xj − xi)/(xj + xi)). Some special cases of these identities are given by S. Okada and T. ..."
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Cited by 9 (4 self)
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We present several identities of Cauchytype determinants and Schurtype Pfaffians involving generalized Vandermonde determinants, which generalize Cauchy’s determinant det(1/(xi + yj)) and Schur’s Pfaffian Pf ((xj − xi)/(xj + xi)). Some special cases of these identities are given by S. Okada and T. Sundquist. As an application, we give a relation for the Littlewood–Richardson coefficients involving a rectangular partition.
Sums of triangular numbers from the Frobenius determinant
"... Abstract. We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact formulas for the number of representations of an arbitra ..."
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Cited by 7 (5 self)
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Abstract. We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact formulas for the number of representations of an arbitrary number as a sum of 4m 2 /d triangles, whenever d  2m, and 4m(m + 1)/d triangles, when d  2m or d  2m + 2. This extends recent results of Getz and Mahlburg, Milne, and Zagier. 1.
CHERN CLASSES OF SCHUBERT CELLS AND VARIETIES
, 2006
"... We give explicit formulas for the ChernSchwartzMacPherson classes of all Schubert varieties in the Grassmannian of dplanes in a vector space, and conjecture that these classes are effective. We prove this is the case for (very) small values of d. ..."
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Cited by 7 (0 self)
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We give explicit formulas for the ChernSchwartzMacPherson classes of all Schubert varieties in the Grassmannian of dplanes in a vector space, and conjecture that these classes are effective. We prove this is the case for (very) small values of d.