A generalized Macdonald operator

We present an explicit difference operator diagonalized by the Macdonald polynomials associated with an (arbitrary) admissible pair of irreducible reduced crystallographic root systems. By the duality symmetry, this gives rise to an explicit Pieri formula for the Macdonald polynomials in question. The simplest examples of our construction recover Macdonald's celebrated difference operators and associated Pieri formulas pertaining to the minuscule and quasi-minuscule weights. As further by-products, explicit expansions and Littlewood-Richardson type formulas are obtained for the Macdonald polynomials associated with a special class of small weights.Comment: 11 pages. To appear in Int. Math. Res. Not. IMR

Powers of the Vandermonde determinant, Schur Functions, and recursive formulas

Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, we give recursive formulas for the coefficient of the Schur function s_{\m} in the decomposition of an even power of the Vandermonde determinant in $n + 1$ variables in terms of the coefficient of the Schur function s_{\l} in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of \m is obtained from the Young diagram of \l by adding a tetris type shape to the top or to the left. An extended abstract containing the statement of the results presented here appeared in the Proceedings of FPSAC11Comment: 23 pages; extended abstract appeared in the Proceedings of FPSAC1

Graph hypersurfaces and a dichotomy in the Grothendieck ring

The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of Z generated by the class of a point. Thus, the span of the graph hypersurfaces in the Grothendieck ring is nearly killed by setting the Lefschetz motive L to zero, while it is known that graph hypersurfaces generate the Grothendieck ring over a localization of Z[L] in which L becomes invertible. In particular, this shows that the graph hypersurfaces do not generate the Grothendieck ring prior to localization. The same result yields some information on the mixed Hodge structures of graph hypersurfaces, in the form of a constraint on the terms in their Deligne-Hodge polynomials.Comment: 8 pages, LaTe

On Motives Associated to Graph Polynomials

The appearance of multiple zeta values in anomalous dimensions and $\beta$-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions.Comment: 54

Special Isogenies and Tensor Product Multiplicities

We show that any bijection between two root systems that preserves angles (but not necessarily lengths) gives rise to inequalities relating tensor product multiplicities for the corresponding complex semisimple Lie groups (or Lie algebras). We explain the inequalities in two ways: combinatorially, using Littelmann’s Path Model, and geometrically, using isogenies between algebraic groups defined over an algebraically closed field of positive characteristic

The massless higher-loop two-point function

We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph $G$ to evaluate to multiple zeta values. The criterion depends only on the topology of $G$, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with $6^\mathrm{th}$ roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph $K_{3,4}$ at one edge

Systolic and Diastolic Left Ventricular Mechanics during and after Resistance Exercise

PURPOSE: To improve the current understanding of the impact of resistance exercise on the heart, by examining the acute responses of left ventricular (LV) strain, twist and untwisting rate ('LV mechanics'). METHODS: LV echocardiographic images were recorded in systole and diastole before, during and immediately after (7-12 s) double leg press exercise at two intensities (30% and 60% of maximum strength, 1-repetition-maximum, 1RM). Speckle tracking analysis generated LV strain, twist and untwisting rate data. Additionally, beat-by-beat blood pressure was recorded and systemic vascular resistance (SVR) and LV wall stress were calculated. RESULTS: Responses in both exercise trials were statistically similar (P > 0.05). During effort, stroke volume decreased while SVR and LV wall stress increased (P 0.05). Immediately following exercise, systolic LV mechanics returned to baseline levels (P < 0.05) but LV untwisting rate increased significantly (P < 0.05). CONCLUSIONS: A single, acute bout of double leg-press resistance exercise transiently reduces systolic LV mechanics, but increases diastolic mechanics following exercise, suggesting that resistance exercise has a differential impact on systolic and diastolic heart muscle function. The findings may explain why acute resistance exercise has been associated with reduced stroke volume but chronic exercise training may result in increased LV volumes

Compound basis arising from the basic A1(1)A^{(1)}_{1}-module

A new basis for the polynomial ring of infinitely many variables is constructed which consists of products of Schur functions and Q-functions. The transition matrix from the natural Schur function basis is investigated.Comment: 12 page

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

The effect of an acute bout of resistance exercise on carotid artery strain and strain rate

Arterial wall mechanics likely play an integral role in arterial responses to acute physiological stress. Therefore, this study aimed to determine the impact of low and moderate intensity double-leg press exercise on common carotid artery (CCA) wall mechanics using 2D vascular strain imaging. Short-axis CCA ultrasound images were collected in 15 healthy men (age: 21 ± 3 years; stature: 176.5 ± 6.2 cm; body mass; 80.6 ± 15.3 kg) before, during, and immediately after short-duration isometric double-leg press exercise at 30% and 60% of participants’ one-repetition maximum (1RM: 317 ± 72 kg). Images were analyzed for peak circumferential strain (PCS), peak systolic and diastolic strain rate (S-SR and D-SR) and arterial diameter. Heart rate (HR), systolic and diastolic blood pressure (SBP and DBP) were simultaneously assessed and arterial stiffness indices were calculated post hoc. A two-way repeated measures ANOVA revealed that during isometric contraction, PCS and S-SR decreased significantly (P < 0.01) before increasing significantly above resting levels post-exercise (P < 0.05 and P < 0.01 respectively). Conversely, D-SR was unaltered throughout the protocol (P = 0.25). No significant differences were observed between the 30% and 60% 1RM trials. Multiple regression analysis highlighted that HR, BP and arterial diameter did not fully explain the total variance in PCS, S-SR and D-SR. Acute double-leg press exercise is therefore associated with similar transient changes in CCA wall mechanics at low and moderate intensities. CCA wall mechanics likely provide additional insight into localized intrinsic vascular wall properties beyond current measures of arterial stiffness