Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams

We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries these numbers of matrices are not polynomials in q (Stembridge 98); however, when the set of entries is a Young diagram, the numbers, up to a power of q-1, are polynomials with nonnegative coefficients (Haglund 98). In this paper, we give a number of conditions under which these numbers are polynomials in q, or even polynomials with nonnegative integer coefficients. We extend Haglund's result to complements of skew Young diagrams, and we apply this result to the case when the set of entries is the Rothe diagram of a permutation. In particular, we give a necessary and sufficient condition on the permutation for its Rothe diagram to be the complement of a skew Young diagram up to rearrangement of rows and columns. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl

Scaling limit of vicious walks and two-matrix model

We consider the diffusion scaling limit of the one-dimensional vicious walker model of Fisher and derive a system of nonintersecting Brownian motions. The spatial distribution of $N$ particles is studied and it is described by use of the probability density function of eigenvalues of $N \times N$ Gaussian random matrices. The particle distribution depends on the ratio of the observation time $t$ and the time interval $T$ in which the nonintersecting condition is imposed. As $t/T$ is going on from 0 to 1, there occurs a transition of distribution, which is identified with the transition observed in the two-matrix model of Pandey and Mehta. Despite of the absence of matrix structure in the original vicious walker model, in the diffusion scaling limit, accumulation of contact repulsive interactions realizes the correlated distribution of eigenvalues in the multimatrix model as the particle distribution.Comment: REVTeX4, 12 pages, no figure, minor corrections made for publicatio

Mask formulas for cograssmannian Kazhdan-Lusztig polynomials

We give two contructions of sets of masks on cograssmannian permutations that can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the Iwahori-Hecke algebra. The constructions are respectively based on a formula of Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky. The first construction relies on a basis of the Hecke algebra constructed from principal lower order ideals in Bruhat order and a translation of this basis into sets of masks. The second construction relies on an interpretation of masks as cells of the Bott-Samelson resolution. These constructions give distinct answers to a question of Deodhar.Comment: 43 page

Crystal Graphs and qq-Analogues of Weight Multiplicities for the Root System AnA_n

We give an expression of the $q$-analogues of the multiplicities of weights in irreducible \sl_{n+1}-modules in terms of the geometry of the crystal graph attached to the corresponding U_q(\sl_{n+1})-modules. As an application, we describe multivariate polynomial analogues of the multiplicities of the zero weight, refining Kostant's generalized exponents.Comment: LaTeX file with epic, eepic pictures, 17 pages, November 1994, to appear in Lett. Math. Phy

Fluctuation properties of the TASEP with periodic initial configuration

We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us and here we provide a self-contained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.Comment: 33 pages, 4 figure, LaTeX; We added several references to the general framework and techniques use

Correlation Functions for \beta=1 Ensembles of Matrices of Odd Size

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.Comment: 15 page

Similarity between carotid and coronary artery responses to sympathetic stimulation and the role of alpha-1 receptors in humans.

BACKGROUND: Carotid artery (CCA) dilation occurs in healthy subjects during cold pressor test (CPT), whilst the magnitude of dilation relates to cardiovascular risk. To further explore this phenomena and mechanism, we examined carotid artery responses to different sympathetic tests, with and without α1-receptor blockade, and assessed similarity to these responses between carotid and coronary arteries. METHODS: In randomised order, 10 healthy participants (25{plus minus}3yrs) underwent sympathetic stimulation using CPT (3-minutes left hand immersion in ice-slush) and lower-body negative pressure (LBNP). Before and during sympathetic tests, CCA diameter and velocity (Doppler ultrasound) and left anterior descending (LAD) coronary artery velocity (echocardiography) were recorded across 3-min. Measures were repeated 90-min following selective α1-receptor blockade via oral Prazosin (0.05mg-per-kg bodyweight). RESULTS: CPT significantly increased CCA diameter, LAD maximal velocity and velocity-time integral area-under-the-curve (all P<0.05). In contrast, LBNP resulted in a decrease in CCA diameter, LAD maximal velocity and velocity time integral (VTI, all P<0.05). Following α1-receptor blockade, CCA and LAD velocity responses to CPT were diminished. In contrast, during LBNP (-30mmHg), α1-receptor blockade did not alter CCA or LAD responses. Finally, changes in CCA diameter and LAD VTI-responses to sympathetic stimulation were positively correlated (r=0.66, P<0.01). CONCLUSION: We found distinct carotid artery responses to different tests of sympathetic stimulation, where α1-receptors partly contribute to CPT-induced responses. Finally, we found agreement between carotid and coronary artery responses. These data indicate similarity between carotid and coronary responses to sympathetic tests and the role of α1-receptors that is dependent on the nature of the sympathetic challenge

Affine and toric hyperplane arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure

An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions

A well-known ansatz (`trace method') for soliton solutions turns the equations of the (noncommutative) KP hierarchy, and those of certain extensions, into families of algebraic sum identities. We develop an algebraic formalism, in particular involving a (mixable) shuffle product, to explore their structure. More precisely, we show that the equations of the noncommutative KP hierarchy and its extension (xncKP) in the case of a Moyal-deformed product, as derived in previous work, correspond to identities in this algebra. Furthermore, the Moyal product is replaced by a more general associative product. This leads to a new even more general extension of the noncommutative KP hierarchy. Relations with Rota-Baxter algebras are established.Comment: 59 pages, relative to the second version a few minor corrections, but quite a lot of amendments, to appear in J. Phys.