71,663 research outputs found
On cluster C*-algebras
We introduce a C*-algebra A(x,Q) attached to the cluster x and a quiver Q. If
Q(T) is the quiver coming from a triangulation T of the Riemann surface S with
a finite number of cusps, we prove that the primitive spectrum of A(x,Q(T))
times R is homeomorphic to a generic subset of the Teichmueller space of
surface S. We conclude with an analog of the Tomita-Takesaki theory and the
Connes invariant T(M) for the algebra A(x,Q(T)).Comment: to appear Journal of Function Space
Sweep maps: A continuous family of sorting algorithms
We define a family of maps on lattice paths, called sweep maps, that assign
levels to each step in the path and sort steps according to their level.
Surprisingly, although sweep maps act by sorting, they appear to be bijective
in general. The sweep maps give concise combinatorial formulas for the
q,t-Catalan numbers, the higher q,t-Catalan numbers, the q,t-square numbers,
and many more general polynomials connected to the nabla operator and rational
Catalan combinatorics. We prove that many algorithms that have appeared in the
literature (including maps studied by Andrews, Egge, Gorsky, Haglund, Hanusa,
Jones, Killpatrick, Krattenthaler, Kremer, Orsina, Mazin, Papi, Vaille, and the
present authors) are all special cases of the sweep maps or their inverses. The
sweep maps provide a very simple unifying framework for understanding all of
these algorithms. We explain how inversion of the sweep map (which is an open
problem in general) can be solved in known special cases by finding a "bounce
path" for the lattice paths under consideration. We also define a generalized
sweep map acting on words over arbitrary alphabets with arbitrary weights,
which is also conjectured to be bijective.Comment: 21 pages; full version of FPSAC 2014 extended abstrac
Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces
Quantum analogues of the homogeneous spaces \GL(n)/\SO(n) and
\GL(2n)/\Sp(2n) are introduced. The zonal spherical functions on these
quantum homogeneous spaces are represented by Macdonald's symmetric polynomials
P_{\ld}=P_{\ld}(x_1,\cdots,x_n;q,t) with or
Root numbers and ranks in positive characteristic
For a global field K and an elliptic curve E_eta over K(T), Silverman's
specialization theorem implies that rank(E_eta(K(T))) <= rank(E_t(K)) for all
but finitely many t in P^1(K). If this inequality is strict for all but
finitely many t, the elliptic curve E_eta is said to have elevated rank. All
known examples of elevated rank for K=Q rest on the parity conjecture for
elliptic curves over Q, and the examples are all isotrivial.
Some additional standard conjectures over Q imply that there does not exist a
non-isotrivial elliptic curve over Q(T) with elevated rank. In positive
characteristic, an analogue of one of these additional conjectures is false.
Inspired by this, for the rational function field K = kappa(u) over any finite
field kappa with odd characteristic, we construct an explicit 2-parameter
family E_{c,d} of non-isotrivial elliptic curves over K(T) (depending on
arbitrary c, d in kappa^*) such that, under the parity conjecture, each E_{c,d}
has elevated rank.Comment: 40 pages; last version; to appear in Adv. Mat
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