9 research outputs found
Promotion on oscillating and alternating tableaux and rotation of matchings and permutations
Using Henriques' and Kamnitzer's cactus groups, Sch\"utzenberger's promotion
and evacuation operators on standard Young tableaux can be generalised in a
very natural way to operators acting on highest weight words in tensor products
of crystals.
For the crystals corresponding to the vector representations of the
symplectic groups, we show that Sundaram's map to perfect matchings intertwines
promotion and rotation of the associated chord diagrams, and evacuation and
reversal. We also exhibit a map with similar features for the crystals
corresponding to the adjoint representations of the general linear groups.
We prove these results by applying van Leeuwen's generalisation of Fomin's
local rules for jeu de taquin, connected to the action of the cactus groups by
Lenart, and variants of Fomin's growth diagrams for the Robinson-Schensted
correspondence
Skew characters and cyclic sieving
In 2010, B. Rhoades proved that promotion together with the fake-degree
polynomial associated with rectangular standard Young tableaux give an instance
of the cyclic sieving phenomenon.
We extend this result to all skew standard Young tableaux where the
fake-degree polynomial evaluates to nonnegative integers at roots of unity,
albeit without being able to specify an explicit group action. Put differently,
we determine in which cases a skew character of the symmetric group carries a
permutation representation of the cyclic group.
We use a method proposed by N. Amini and the first author, which amounts to
establishing a bound on the number of border-strip tableaux of skew shape.
Finally, we apply our results to the invariant theory of tensor powers of the
adjoint representation of the general linear group. In particular, we prove the
existence of a bijection between permutations and J. Stembridge's alternating
tableaux, which intertwines rotation and promotion
Promotion and growth diagrams for fans of Dyck paths and vacillating tableaux
We construct an injection from the set of -fans of Dyck paths (resp.
vacillation tableaux) of length into the set of chord diagrams on
that intertwines promotion and rotation. This is done in two different ways,
namely as fillings of promotion--evacuation diagrams and in terms of Fomin
growth diagrams. Our analysis uses the fact that -fans of Dyck paths and
vacillating tableaux can be viewed as highest weight elements of weight zero in
crystals of type and , respectively, which in turn can be analyzed
using virtual crystals. On the level of Fomin growth diagrams, the
virtualization process corresponds to Krattenthaler's blow up construction. One
of the motivations for finding rotation invariant diagrammatic bases such as
chord diagrams is the cyclic sieving phenomenon. Indeed, we give a cyclic
sieving phenomenon on -fans of Dyck paths and vacillating tableaux using the
promotion action.Comment: 40 pages, 13 figure
Promotion permutations for tableaux
In our companion paper, we develop a new -web basis. Basis elements are
given by certain planar graphs and are constructed so that important algebraic
operations can be performed diagrammatically. A guiding principle behind our
construction is that the long cycle should
act by rotation of webs. Moreover, the bijection between webs and tableaux
should intertwine rotation with the promotion action on tableaux.
In this paper, we develop necessary notions of promotion permutations and
promotion matrices, which are new even for standard tableaux. To support
inductive arguments in the companion paper, we must however work in the more
general setting of fluctuating tableaux, which we introduce and which subsumes
many classes of tableaux that have been previously studied, including
(generalized) oscillating, vacillating, rational, alternating, and
(semi)standard tableaux. Therefore, we also give here a full development of the
basic combinatorics and representation theory of fluctuating tableaux.Comment: 42 pages, 8 figure
A refinement of the Murnaghan-Nakayama rule by descents for border strip tableaux
Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer (1974) and James & Kerber (1984) imply that, mysteriously, its evaluation at a -th primitive root of unity yields the number of border strip tableaux with all strips of size , up to sign. This is essentially the special case of the Murnaghan-Nakayama rule for evaluating an irreducible character of the symmetric group at a rectangular partition. We refine this result to standard Young tableaux and border strip tableaux with a given number of descents. To do so, we introduce a new statistic for border strip tableaux, extending the classical definition of descents in standard Young tableaux. Curiously, it turns out that our new statistic is very closely related to a descent set for tuples of standard Young tableaux appearing in the quasisymmetric expansion of LLT polynomials given by Haglund, Haiman and Loehr (2005).Mathematics Subject Classifications: 05A19, 05E10Keywords: Border strip tableaux, descents, Murnaghan-Nakayama rule, fake degre
Kristalline Graphen, Promotion, Evacuation und Kaktusgruppen
Abweichender Titel nach Übersetzung der Verfasserin/des VerfassersUsing Henriques' and Kamnitzer's cactus groups, Schützenberger's promotion and evacuation operators on standard Young tableaux can be generalised in a very natural way to operators acting on highest weight words in tensor products of crystals. For the crystals corresponding to the vector representations of the symplectic groups, we show that Sundaram's map to perfect matchings intertwines promotion and rotation of the associated chord diagrams, and evacuation and reversal. We also exhibit a map with similar features for the crystals corresponding to the adjoint representations of the general linear groups. We prove these results by applying van Leeuwen's generalisation of Fomin's local rules for jeu de taquin, connected to the action of the cactus groups by Lenart, and variants of Fomin's growth diagrams for the Robinson-Schensted correspondence. This work is based on a joint research project with Martin Rubey and Bruce W. Westbury. In chapter 1 we give a general introduction and state related work. Chapter 2 connects the algebraic world of representations with combinatorics and we present our findings in chapter 3. In chapter 4 we define promotion and evacuation as actions of certain elements of a cactus group and state local rules for algorithmically calculating these actions. The local rules are strongly related to the rules of our growth diagram bijections from chapter 5. The last chapter 6 is meant for proofs only. Chapters 1, 3, 4, 5 and 6 are also published separately as a joint paper.6
Recommended from our members
Promotion and growth diagrams for fans of Dyck paths and vacillating tableaux
Rotation-invariant web bases from hourglass plabic graphs
Webs give a diagrammatic calculus for spaces of tensor invariants. We
introduce hourglass plabic graphs as a new avatar of webs, and use these to
give the first rotation-invariant -web basis, a
long-sought object. The characterization of our basis webs relies on the
combinatorics of these new plabic graphs and associated configurations of a
symmetrized six-vertex model. We give growth rules, based on a novel
crystal-theoretic technique, for generating our basis webs from tableaux and we
use skein relations to give an algorithm for expressing arbitrary webs in the
basis. We also discuss how previously known rotation-invariant web bases can be
unified in our framework of hourglass plabic graphs.Comment: 60 pages, 38 figure