2,497 research outputs found
The Robinson-Schensted Correspondence and -web Bases
We study natural bases for two constructions of the irreducible
representation of the symmetric group corresponding to : the {\em
reduced web} basis associated to Kuperberg's combinatorial description of the
spider category; and the {\em left cell basis} for the left cell construction
of Kazhdan and Lusztig. In the case of , the spider category is the
Temperley-Lieb category; reduced webs correspond to planar matchings, which are
equivalent to left cell bases. This paper compares the images of these bases
under classical maps: the {\em Robinson-Schensted algorithm} between
permutations and Young tableaux and {\em Khovanov-Kuperberg's bijection}
between Young tableaux and reduced webs.
One main result uses Vogan's generalized -invariant to uncover a close
structural relationship between the web basis and the left cell basis.
Intuitively, generalized -invariants refine the data of the inversion set
of a permutation. We define generalized -invariants intrinsically for
Kazhdan-Lusztig left cell basis elements and for webs. We then show that the
generalized -invariant is preserved by these classical maps. Thus, our
result allows one to interpret Khovanov-Kuperberg's bijection as an analogue of
the Robinson-Schensted correspondence.
Despite all of this, our second main result proves that the reduced web and
left cell bases are inequivalent; that is, these bijections are not
-equivariant maps.Comment: 34 pages, 23 figures, minor corrections and revisions in version
Normal 6-edge-colorings of some bridgeless cubic graphs
In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of
colors assigned to the edge and the four edges adjacent it, has exactly five or
exactly three distinct colors, respectively. An edge is normal in an
edge-coloring if it is rich or poor in this coloring. A normal
-edge-coloring of a cubic graph is an edge-coloring with colors such
that each edge of the graph is normal. We denote by the smallest
, for which admits a normal -edge-coloring. Normal edge-colorings
were introduced by Jaeger in order to study his well-known Petersen Coloring
Conjecture. It is known that proving for every bridgeless
cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover,
Jaeger was able to show that it implies classical conjectures like Cycle Double
Cover Conjecture and Berge-Fulkerson Conjecture. Recently, two of the authors
were able to show that any simple cubic graph admits a normal
-edge-coloring, and this result is best possible. In the present paper, we
show that any claw-free bridgeless cubic graph, permutation snark, tree-like
snark admits a normal -edge-coloring. Finally, we show that any bridgeless
cubic graph admits a -edge-coloring such that at least edges of are normal.Comment: 17 pages, 11 figures. arXiv admin note: text overlap with
arXiv:1804.0944
Is the five-flow conjecture almost false?
The number of nowhere zero Z_Q flows on a graph G can be shown to be a
polynomial in Q, defining the flow polynomial \Phi_G(Q). According to Tutte's
five-flow conjecture, \Phi_G(5) > 0 for any bridgeless G.A conjecture by Welsh
that \Phi_G(Q) has no real roots for Q \in (4,\infty) was recently disproved by
Haggard, Pearce and Royle. These authors conjectured the absence of roots for Q
\in [5,\infty). We study the real roots of \Phi_G(Q) for a family of non-planar
cubic graphs known as generalised Petersen graphs G(m,k). We show that the
modified conjecture on real flow roots is also false, by exhibiting infinitely
many real flow roots Q>5 within the class G(nk,k). In particular, we compute
explicitly the flow polynomial of G(119,7), showing that it has real roots at
Q\approx 5.0000197675 and Q\approx 5.1653424423. We moreover prove that the
graph families G(6n,6) and G(7n,7) possess real flow roots that accumulate at
Q=5 as n\to\infty (in the latter case from above and below); and that
Q_c(7)\approx 5.2352605291 is an accumulation point of real zeros of the flow
polynomials for G(7n,7) as n\to\infty.Comment: 44 pages (LaTeX2e). Includes tex file, three sty files, and a
mathematica script polyG119_7.m. Many improvements from version 3, in
particular Sections 3 and 4 have been mostly re-writen, and Sections 7 and 8
have been eliminated. (This material can now be found in arXiv:1303.5210.)
Final version published in J. Combin. Theory
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