Symmetries of the Young lattice and abelian ideals of Borel

47 Pages·2013·0.59 MB·English

by  Paolo Papi

Symmetries of the Young lattice and abelian ideals
of Borel subalgebras
Paolo Papi
SapienzaUniversit`adiRoma
joint work with P. Cellini and P. M¨oseneder Frajria
PaoloPapi SymmetriesoftheYounglattice
Starting point: a theorem of Suter
(Europ. J. Comb., 2002)
For a positive integer n let Y be the Hasse graph for the subposet
n
Y of the Young lattice corresponding to subdiagrams of the
n
staircase diagram for the partition (n−1,n−2,...,1) with hook
length ≤ n−1 .
Theorem
If n ≥ 3, the dihedral group of order 2n acts faithfully on the
(undirected) graph Y .
n
PaoloPapi SymmetriesoftheYounglattice
Example
Figure: The Hasse diagram of Y and its underlying graph.
5
PaoloPapi SymmetriesoftheYounglattice
σ is given by the following tricky
n
procedure.
One has to verify that σ acts on Y , maps edges to edges, and
n n
has order n.
Idea of the proof
It suffices to makes explicit the action of generators
σ ,τ,o(σ ) = n,o(τ) = 2
n n
of the dihedral group.
PaoloPapi SymmetriesoftheYounglattice
σ is given by the following tricky procedure.
n
One has to verify that σ acts on Y , maps edges to edges, and
n n
has order n.
Idea of the proof
τ is simply the transposition of the diagram.
PaoloPapi SymmetriesoftheYounglattice
One has to verify that σ acts on Y , maps edges to edges, and
n n
has order n.
Idea of the proof
σ is given by the following tricky procedure.
n
PaoloPapi SymmetriesoftheYounglattice
One has to verify that σ acts on Y , maps edges to edges, and
n n
has order n.
Idea of the proof
σ is given by the following tricky procedure.
n
PaoloPapi SymmetriesoftheYounglattice
Idea of the proof
σ is given by the following tricky procedure.
n
One has to verify that σ acts on Y , maps edges to edges, and
n n
has order n.
PaoloPapi SymmetriesoftheYounglattice
Remark 1
This result is conceptually due to Suter; details can be found in
our (Cellini-M¨oseneder-P.) paper arXiv:1301.2548, to appear in
Journal of Lie Theory
Remark 2
Suter’s dihedral symmetry got recently much attention: see e.g.
Berg,C.,ZabrockiM.,Symmetriesonthelatticeofk-boundedpartitions,
arXiv:1111.2783v2.
SuterR.,Youngslatticeanddihedralsymmetriesrevisited: M¨obiusstrips&metric
geometry,arXiv:1212.4463v1
ThomasH.,N.Willliams,Cyclicsymmetryofthescaledsimplex,arXiv:1207.5240v1
This talk
Goal of this talk
We want to provide a representation-theoretic interpretation of this
result via the theory of abelian ideals of Borel subalgebras.
PaoloPapi SymmetriesoftheYounglattice
Remark 2
Suter’s dihedral symmetry got recently much attention: see e.g.
Berg,C.,ZabrockiM.,Symmetriesonthelatticeofk-boundedpartitions,
arXiv:1111.2783v2.
SuterR.,Youngslatticeanddihedralsymmetriesrevisited: M¨obiusstrips&metric
geometry,arXiv:1212.4463v1
ThomasH.,N.Willliams,Cyclicsymmetryofthescaledsimplex,arXiv:1207.5240v1
This talk
Goal of this talk
We want to provide a representation-theoretic interpretation of this
result via the theory of abelian ideals of Borel subalgebras.
Remark 1
This result is conceptually due to Suter; details can be found in
our (Cellini-M¨oseneder-P.) paper arXiv:1301.2548, to appear in
Journal of Lie Theory
PaoloPapi SymmetriesoftheYounglattice