The symbolical and cancellation-free formulae for Schur elements

In this paper we give the symbolical formula and cancellation-free formula for the Schur elements associated to the simple modules of the degenerate cyclotomic Hecke algebras. As some applications, we show that the Schur elements are symmetric polynomials with rational integer coefficients and give a different proof of Ariki–Mathas–Rui’s criterion on the semisimplicity of the degenerate cyclotomic Hecke algebras.     

A Matrix Identity on the Schur Complement

This article presents a matrix identity on the Schur complement along with various applications. In particular, it gives a simple and transparent proof for the Crabtree–Haynsworth quotient formula for the Schur complement. Although its proof is straightforward, the identity yields a number of important results that appear to be unrelated.     

A super Frobenius formula for the characters of Iwahori–Hecke algebras

In this article, we establish a super Frobenius formula for the characters of Iwahori–Hecke algebras. We define Hall–Littlewood supersymmetric functions in a standard manner to make supersymmetric functions from symmetric functions, and give some properties of supersymmetric functions. Based on Schur–Weyl reciprocity between Iwahori–Hecke algebras and the general quantum super algebras, which was obtained in Mitsuhashi [H. Mitsuhashi, Schur–Weyl reciprocity between the quantum superalgebra and the Iwahori–Hecke algebra, Algeb. Represent. Theor. 9 (2006), pp. 309–322.], we derive that the Hall–Littlewood supersymmetric functions, up to constant, generates the values of the irreducible characters of Iwahori–Hecke algebras at the elements corresponding to cycle permutations. Our formula in this article includes both the ordinary quantum case that was obtained in Ram [A. Ram, A Frobenius formula for the characters of the Hecke algebra, Invent. Math. 106 (1991), pp. 461–488.] and the classical super case.     

Multiplicative Lie algebras and Schur multiplier

In this paper, we attempt to study the structure of multiplicative Lie algebras, the theory of extensions, the second cohomology groups of multiplicative Lie algebras, and in turn the Schur multipliers. The Schur–Hopf formula is established for multiplicative Lie algebras. We also introduce the group of nontrivial relations satisfied by the Lie product in a multiplicative Lie algebra, and study it as a functor arising from the presentations of multiplicative Lie algebras. Some applications in K-theory are also discussed.     

Twisted exponents and twisted Frobenius–Schur indicators for Hopf algebras

Classically, the exponent of a group is the least common multiple of the orders of its elements. This notion was generalized by Etingof and Gelaki to Hopf algebras. Kashina, Sommerhäuser, and Zhu later observed that there is a strong connection between exponents and Frobenius–Schur indicators. In this article, we introduce the notion of twisted exponents and show there is a similar relationship between the twisted exponent and the twisted Frobenius–Schur indicators defined in previous work of the authors. In particular, we exhibit a new formula for the twisted indicators and use it to prove periodicity and rationality statements.     

Jucys–Murphy elements and a presentation for partition algebras

We give a new presentation for the partition algebras. This presentation was discovered in the course of establishing an inductive formula for the partition algebra Jucys–Murphy elements defined by Halverson and Ram (Eur. J. Comb. 26:869–921, 2005). Using Schur–Weyl duality we show that our recursive formula and the original definition of Jucys–Murphy elements given by Halverson and Ram are equivalent. The new presentation and inductive formula for the partition algebra Jucys–Murphy elements given in this paper are used to construct the seminormal representations for the partition algebras in a separate paper.     

The matricial Schur problem in both nondegenerate and degenerate cases

The principal object of this paper is to present a new approach simultaneously to both nondegenerate and degenerate cases of the matricial Schur problem. This approach is based on an analysis of the central matrixvalued Schur functions which was started in [24]–[26] and then continued in [27]. In the nondegenerate situation we will see that the parametrization of the solution set obtained here coincides with the well‐known formula of D. Z. Arov and M. G. Kre?n for that case (see [1]). Furthermore, we give some characterizations of the situation that the matricial Schur problem has a unique solution (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pieri’s formula for generalized Schur polynomials

Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutation relation of generalized Schur operators implies Pieri's formula for generalized Schur polynomials.     

Moments of the Hermitian Matrix Jacobi Process

In this paper, we compute the expectation of traces of powers of the Hermitian matrix Jacobi process for a large enough but fixed size. To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy–Binet formula in order to determine the partitions having nonzero contributions after integration. It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized beta distribution. For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity.     

Remarks on the Schur complement

First, a generalization of the Schur determinantal formula is given. Using properties of quasidirect sums of matrices, a new characterization of the Schur complement is proved.     

Combinatorial Formula for Factorial Schur Q-Functions

We study an interpolation analogue of the Schur Q-functions, the factorial Schur Q-functions. We obtain an expression of factorial Schur Q-functions in terms of shifted tableaux (combinatorial formula). Bibliography: 13 titles.     

Interpolation Analogs of Schur <Emphasis Type="Italic">Q</Emphasis>-Functions

We introduce interpolation analogs of the Schur Q-functions — the multiparameter Schur Q-functions. We obtain for them several results: a combinatorial formula, generating functions for one-row and two-row functions, vanishing and characterization properties, a Pieri-type formula, a Nimmo-type formula (a quotient of two Pfaffians), a Giambelli-Schur-type Pfaffian formula, a determinantal formula for the transition coefficients between multiparameter Schur Q-functions with different parameters. We write an explicit Pfaffian expression for the dimension of a skew shifted Young diagram. This paper is a continuation of the author's paper math. CO/0303169 and a partial projective analog of the paper q-alg/9605042 by A. Okounkov and G. Olshanski and the paper math. CO/0110077 by G. Olshanski, A. Regev, and A. Vershik. Bibliography: 36 titles. __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2004, pp. 99–119.     

Haglund’s conjecture on 3-column Macdonald polynomials

We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund (Proc Natl Acad Sci USA 101(46):16127–16131, 2004). The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam’s (Eur J Combin 29(1):343–359, 2008) algebra of ribbon Schur operators. Combining this result with the expression of Haglund et al. (J Am Math Soc 18(3):735–761, 2005) for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Macdonald polynomials indexed by a shape with 3 columns.     

Schur-Finiteness and Endomorphisms Universally of Trace Zero Via Certain Trace Relations

We provide a sufficient condition that ensures the nilpotency of endomorphisms universally of trace zero of Schur-finite objects in a category of homological type, i.e., a ?-linear ?-category with a tensor functor to super vector spaces. This generalizes previous results about finite-dimensional objects, in particular by Kimura in the category of motives. We also present some facts which suggest that this might be the best generalization possible of this line of proof. To get the result we prove an identity of trace relations on super vector spaces which has an independent interest in the field of combinatorics. Our main tool is Berele–Regev's theory of Hook Schur functions. We use their generalization of the classic Schur–Weyl duality to the “super” case, together with their factorization formula.     

The generalized Schur complement in group inverses and (k 1)-potent matrices

In this article, two facts related to the generalized Schur complement are studied. The first one is to find necessary and sufficient conditions to characterize when the group inverse of a partitioned matrix can be expressed in the Schur form. The other one is to develop a formula for any power of the generalized Schur complement of an idempotent partitioned matrix and then to characterize when this generalized Schur complement is a (k+1)-potent matrix. In addition, some spectral theory related to this complement is analyzed.     

Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities

The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [Ishikawa, Wakayama, Minor summation formula of Pfaffians, Linear and Multilinear Algebra 39 (1995) 285-305] based on the lattice path method. The second aim is to study applications of the minor summation formula for obtaining several identities. Especially, a simple proof of Kawanaka's formula concerning a q-series identity involving the Schur functions [Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999) 157-176] and of the identity in [Kawanaka, A q-Cauchy identity involving Schur functions and imprimitive complex reflection groups, Osaka J. Math. 38 (2001) 775-810] which is regarded as a determinant version of the previous one are given.     

Spin invariant theory for the symmetric group

We formulate a theory of invariants for the spin symmetric group in some suitable modules which involve the polynomial and exterior algebras. We solve the corresponding graded multiplicity problem in terms of specializations of the Schur Q-functions and a shifted q-hook formula. In addition, we provide a bijective proof for a formula of the principal specialization of the Schur Q-functions.     

A Matrix Identity on the Schur Complement

This article presents a matrix identity on the Schur complement along with various applications. In particular, it gives a simple and transparent proof for the Crabtree-Haynsworth quotient formula for the Schur complement. Although its proof is straightforward, the identity yields a number of important results that appear to be unrelated.     

Cylindric skew Schur functions

Cylindric skew Schur functions, which are a generalisation of skew Schur functions, arise naturally in the study of P-partitions. Also, recent work of A. Postnikov shows they have a strong connection with a problem of considerable current interest: that of finding a combinatorial proof of the non-negativity of the 3-point Gromov-Witten invariants. After explaining these motivations, we study cylindric skew Schur functions from the point of view of Schur-positivity. Using a result of I. Gessel and C. Krattenthaler, we generalise a formula of A. Bertram, I. Ciocan-Fontanine and W. Fulton, thus giving an expansion of an arbitrary cylindric skew Schur function in terms of skew Schur functions. While we show that no non-trivial cylindric skew Schur functions are Schur-positive, we conjecture that this can be reconciled using the new concept of cylindric Schur-positivity.     

Valiron's theorem in the unit ball and spectra of composition operators

We prove a version of Valiron's conjugacy theorem for Schur class mappings of the unit ball of CN. As an application we obtain a formula for the spectral radius of composition operators on the ball with Schur class symbols.