A summation formula for Macdonald polynomials

We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases $t=1$ and $q=0$, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and $q$-Whittaker polynomials.Comment: 8 page

Matrix product and sum rule for Macdonald polynomials

We present a new, explicit sum formula for symmetric Macdonald polynomials $P_\lambda$ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov--Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang--Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one.Comment: 11 pages, extended abstract submission to FPSA

Matrix product formula for Macdonald polynomials

We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik--Zamolodchikov equations, which arise by considering representations of the Zamolodchikov--Faddeev and Yang--Baxter algebras in terms of $t$-deformed bosonic operators. These solutions form a basis of the ring of polynomials in $n$ variables, whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalisations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at $q=1$.Comment: 27 pages; typos corrected, references added and some better conventions adopted in v

Speak Through Suiboku

H you should one day settle down with a fude or two, a suzuri, some kami, and a piece of sumi, you might stumble onto a whole new world of beauty and enjoyment. These strange-sounding things are the instruments of suiboku, an oriental art technique which is centuries old

Gay Gifts Inside and Out

Tie on a shining bow or tack on a shimmering bangle when you\u27re wrapping packages for those special people this coming Christmas, and watch the shine in their eyes. The joy both you and they will know through the time and care you take will more than match your extra efforts. So gather up your supplies, and lock yourself away with your imagination

Uniquely Yours

Did you ever have that feeling that the Christmas cards you sent all had the wrong addresses on them and were being returned - and actually, it was just that your friends had chosen the same box of greetings as you had - and you were receiving the same type that you had mailed

Capital Expenditure Decisions and the Role of the Not-for-Profit Hospital: An Application of a Social Goods Model

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/68370/2/10.1177_107755879004700404.pd

Transition probability and total crossing events in the multi-species asymmetric exclusion process

We present explicit formulas for total crossing events in the multi-species asymmetric exclusion process ($r$-ASEP) with underlying $U_q(\widehat{\mathfrak{sl}}_{r+1})$ symmetry. In the case of the two-species TASEP these can be derived using an explicit expression for the general transition probability on $\mathbb{Z}$ in terms of a multiple contour integral derived from a nested Bethe ansatz approach. For the general $r$-ASEP we employ a vertex model approach within which the probability of total crossing can be derived from partial symmetrization of an explicit high rank rainbow partition function. In the case of $r$-TASEP, the total crossing probability can be show to reduce to a multiple integral over the product of $r$ determinants. For $2$-TASEP we additionally derive convenient formulas for cumulative total crossing probabilities using Bernoulli-step initial conditions for particles of type 2 and type 1 respectively.Comment: 41 pages, 4 figure