Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra

We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an ordinary skew diagram of two partitions, viewed as a subset of the two-dimensional integer lattice, by the period vector (n,-k). Imposing a periodicity condition one defines cylindric skew tableaux as a map from the periodically continued skew diagram into the integers. The resulting cylindric Macdonald functions appear in the coproduct of a commutative Frobenius algebra, which is a particular quotient of the spherical Hecke algebra. We realise this Frobenius algebra as a commutative subalgebra in the endomorphisms over a Kirillov-Reshetikhin module of the quantum affine sl(n) algebra. Acting with special elements of this subalgebra, which are noncommutative analogues of Macdonald polynomials, on a highest weight vector, one obtains Lusztig's canonical basis. In the limit q=0, one recovers the sl(n) Verlinde algebra, i.e. the structure constants of the Frobenius algebra become the WZNW fusion coefficients which are known to be dimensions of moduli spaces of generalized theta-functions and multiplicities of tilting modules of quantum groups at roots of unity. Further motivation comes from exactly solvable lattice models in statistical mechanics: the cylindric Macdonald functions arise as partition functions of so-called vertex models obtained from solutions to the quantum Yang-Baxter equation. We show this by stating explicit bijections between cylindric tableaux and lattice configurations of non-intersecting paths. Using the algebraic Bethe ansatz the idempotents of the Frobenius algebra are computed.Comment: 77 pages, 12 figures; v3: some minor typos corrected and title slightly changed. Version to appear in Comm. Math. Phy

Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation

Lepton and quark flavor invariants are studied, both in the Standard Model with a dimension five Majorana neutrino mass operator, and in the seesaw model. The ring of invariants in the lepton sector is highly non-trivial, with non-linear relations among the basic invariants. The invariants are classified for the Standard Model with two and three generations, and for the seesaw model with two generations, and the Hilbert series is computed. The seesaw model with three generations proved computationally too difficult for a complete solution. We give an invariant definition of the CP-violating angle theta in the electroweak sector

Plethysm and lattice point counting

We apply lattice point counting methods to compute the multiplicities in the plethysm of $GL(n)$. Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition $\mu$ of 3,4, or 5 we obtain an explicit formula in $\lambda$ and $k$ for the multiplicity of $S^\lambda$ in $S^\mu(S^k)$.Comment: 25 pages including appendix, 1 figure, computational results and code available at http://thomas-kahle.de/plethysm.html, v2: various improvements, v3: final version appeared in JFoC

Representations of Rational Cherednik algebras with minimal support and torus knots

We obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type A_{n-1} for c=m/n, and an expression of its quasispherical part (i.e., the isotypic part of "hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Etingof to give explicit formulas for the characters of the irreducible equivariant D-modules on the nilpotent cone for SL_m. Our third result is the construction of the Koszul-BGG complex for the rational Cherednik algebra, which generalizes the construction of the Koszul-BGG resolution by Berest-Etingof-Ginzburg and Gordon, and the calculation of its homology in type A. We also show in type A that the differentials in the Koszul-BGG complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the Cherednik algebra. Finally, our fourth result is the symmetry theorem, which identifies the quasispherical components in the representations with minimal support over the rational Cherednik algebras H_{m/n}(S_n) and H_{n/m}(S_m). In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry has a natural interpretation in terms of invariants of torus knots.Comment: 45 pages, latex; the new version contains a new subsection 3.4 on the Cohen-Macaulay property of subspace arrangements and a strengthened version of Theorem 1.

Invariants, Kronecker Products, and Combinatorics of Some Remarkable Diophantine Systems (Extended Version)

This work lies across three areas (in the title) of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link consists of a single formal power series with a multifaced interpretation. The deeper exploration of this link yielded results as well as methods for solving some numerical problems in each of these separate areas.Comment: 33 pages, 5 figure