Characterization of the Graded Betti Numbers for Almost Complete Intersections

Using standard mapping cone procedure of the inclusion map of an ideal generated by a suitable regular sequence inside a Gorenstein ideal and by a deep analysis of possible cancellations, we give a numerical characterization of the graded Betti sequences admissible for almost complete intersections of height three.     

Minor summation formula and a proof of Stanley’s open problem

In the open problem session of the FPSAC’03, R.P. Stanley gave an open problem about a certain sum of the Schur functions. The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear Algebra 39:285–305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist’s Pfaffian identities (J. Algebr. Comb. 5:135–148, 1996). Then we give a proof of Stanley’s open problem in Sect. 4. At the end of this paper we present certain corollaries obtained from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize Stanley’s open problem.      

Dimension and enumeration of primitive ideals in quantum algebras

In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of 2×n quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thought of as giving an enumeration of the points that are invariant under the induced action of the torus in the “variety of 2×n quantum matrices”. The first author thanks NSERC for its generous support. This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme held at the University of Edinburgh, by a Marie Curie European Reintegration Grant within the 7th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.     

Enumeration of domino tilings of an Aztec rectangle with boundary defects

In this paper we enumerate domino tilings of an Aztec rectangle with arbitrary defects of size one on all boundary sides. This result extends previous work by different authors: Mills–Robbins–Rumsey and Elkies–Kuperberg–Larsen–Propp. We use the method of graphical condensation developed by Kuo and generalized by Ciucu, to prove our results; a common generalization of both Kuo's and Ciucu's result is also presented here.     

Generalizing the KP Hierarchies: Pfaffian Hierarchies

We derive the Pfaffian analogues of the equations in the single-component KP hierarchies and the modified KP hierarchies and present an example of a system derived by reduction of some of the equations in these Pfaffianized hierarchies.     

The Andrews-Stanley partition function and Al-Salam-Chihara polynomials

For any partition λ let ω(λ) denote the four parameter weight

ω(λ)=a^{∑_i≥1⌈λ_2i−1/2⌉}b^{∑_i≥1⌊λ_2i−1/2⌋}c^{∑_i≥1⌈λ_2i/2⌉}d^{∑_i≥1⌊λ_2i/2⌋},     

Pfaffians and representations of the symmetric group

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 Pliicker 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.