57,636 research outputs found
The diameter of type D associahedra and the non-leaving-face property
Generalized associahedra were introduced by S. Fomin and A. Zelevinsky in
connection to finite type cluster algebras. Following recent work of L. Pournin
in types and , this paper focuses on geodesic properties of generalized
associahedra. We prove that the graph diameter of the -dimensional
associahedron of type is precisely for all greater than .
Furthermore, we show that all type associahedra have the non-leaving-face
property, that is, any geodesic connecting two vertices in the graph of the
polytope stays in the minimal face containing both. This property was already
proven by D. Sleator, R. Tarjan and W. Thurston for associahedra of type .
In contrast, we present relevant examples related to the associahedron that do
not always satisfy this property.Comment: 18 pages, 14 figures. Version 3: improved presentation,
simplification of Section 4.1. Final versio
Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule
We prove that the sum of entries of the suitably normalized groundstate
vector of the O(1) loop model with periodic boundary conditions on a periodic
strip of size 2n is equal to the total number of n x n alternating sign
matrices. This is done by identifying the state sum of a multi-parameter
inhomogeneous version of the O(1) model with the partition function of the
inhomogeneous six-vertex model on a n x n square grid with domain wall boundary
conditions.Comment: 30 pages. v2: Eq. (3.38) corrected. v3: title changed, references
added. v4: q and q^{-1} switched to conform to standard convention
Efficient Semidefinite Spectral Clustering via Lagrange Duality
We propose an efficient approach to semidefinite spectral clustering (SSC),
which addresses the Frobenius normalization with the positive semidefinite
(p.s.d.) constraint for spectral clustering. Compared with the original
Frobenius norm approximation based algorithm, the proposed algorithm can more
accurately find the closest doubly stochastic approximation to the affinity
matrix by considering the p.s.d. constraint. In this paper, SSC is formulated
as a semidefinite programming (SDP) problem. In order to solve the high
computational complexity of SDP, we present a dual algorithm based on the
Lagrange dual formalization. Two versions of the proposed algorithm are
proffered: one with less memory usage and the other with faster convergence
rate. The proposed algorithm has much lower time complexity than that of the
standard interior-point based SDP solvers. Experimental results on both UCI
data sets and real-world image data sets demonstrate that 1) compared with the
state-of-the-art spectral clustering methods, the proposed algorithm achieves
better clustering performance; and 2) our algorithm is much more efficient and
can solve larger-scale SSC problems than those standard interior-point SDP
solvers.Comment: 13 page
Jack polynomials and orientability generating series of maps
We study Jack characters, which are the coefficients of the power-sum
expansion of Jack symmetric functions with a suitable normalization. These
quantities have been introduced by Lassalle who formulated some challenging
conjectures about them. We conjecture existence of a weight on non-oriented
maps (i.e., graphs drawn on non-oriented surfaces) which allows to express any
given Jack character as a weighted sum of some simple functions indexed by
maps. We provide a candidate for this weight which gives a positive answer to
our conjecture in some, but unfortunately not all, cases. In particular, it
gives a positive answer for Jack characters specialized on Young diagrams of
rectangular shape. This candidate weight attempts to measure, in a sense, the
non-orientability of a given map.Comment: v2: change of title, substantial changes of the content v3:
substantial changes in the presentatio
Tropical curves, graph complexes, and top weight cohomology of M_g
We study the topology of a space parametrizing stable tropical curves of
genus g with volume 1, showing that its reduced rational homology is
canonically identified with both the top weight cohomology of M_g and also with
the genus g part of the homology of Kontsevich's graph complex. Using a theorem
of Willwacher relating this graph complex to the Grothendieck-Teichmueller Lie
algebra, we deduce that H^{4g-6}(M_g;Q) is nonzero for g=3, g=5, and g at least
7. This disproves a recent conjecture of Church, Farb, and Putman as well as an
older, more general conjecture of Kontsevich. We also give an independent proof
of another theorem of Willwacher, that homology of the graph complex vanishes
in negative degrees.Comment: 31 pages. v2: streamlined exposition. Final version, to appear in J.
Amer. Math. So
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