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 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

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

Is Language the Ultimate Artefact?

Andy Clark has argued that language is “in many ways the ultimate artifact” (Clark 1997, p.218). Fuelling this conclusion is a view according to which the human brain is essentially no more than a patterncompleting device, while language is an external resource which is adaptively fitted to the human brain in such a way that it enables that brain to exceed its unaided (pattern-completing) cognitive capacities, in much the same way as a pair of scissors enables us to “exploit our basic manipulative capacities to fulfill new ends” (Clark 1997, pp.193-4). How should we respond to this bold reconceptualization of our linguistic abilities? First we need to understand it properly. So I begin by identifying and unpacking (and making a small “Heideggerian” amendment to) Clark’s main language-specific claims. That done I take a step back. Clark’s approach to language is generated from a theoretical perspective which sees cognition as distributed over brain, body, and world. So I continue my investigation of Clark’s incursion into linguistic territory by uncovering and illustrating those key ideas from the overall distributed cognition research programme which are particularly relevant in the present context. I then use this analysis as a spring-board from which to examine a crucial issue that arises for Clark’s account of language, namely linguistic inner rehearsal. I argue that while there is much to recommend in Clark’s treatment of this issue, some significant difficulties remain to be overcome. Via this critique of Clark’s position, alongside some proposals for how the revealed problems might be addressed, I hope to edge us that bit closer to a full understanding of our linguistic abilities