Orbits of Plane Partitions of Exceptional Lie Type

For each minuscule flag variety $X$, there is a corresponding minuscule poset, describing its Schubert decomposition. We study an action on plane partitions over such posets, introduced by P. Cameron and D. Fon-der-Flaass (1995). For plane partitions of height at most $2$, D. Rush and X. Shi (2013) proved an instance of the cyclic sieving phenomenon, completely describing the orbit structure of this action. They noted their result does not extend to greater heights in general; however, when $X$ is one of the two minuscule flag varieties of exceptional Lie type $E$, they conjectured explicit instances of cyclic sieving for all heights. We prove their conjecture in the case that $X$ is the Cayley-Moufang plane of type $E_6$. For the other exceptional minuscule flag variety, the Freudenthal variety of type $E_7$, we establish their conjecture for heights at most $4$, but show that it fails generally. We further give a new proof of an unpublished cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height in the case $X$ is an even-dimensional quadric hypersurface. Our argument uses ideas of K. Dilks, O. Pechenik, and J. Striker (2017) to relate the action on plane partitions to combinatorics derived from $K$-theoretic Schubert calculus.Comment: 25 pages, 7 figures, 3 tables. Section 5 rewritten and simplifie

Series of nilpotent orbits

We organize the nilpotent orbits in the exceptional complex Lie algebras into series using the triality model and show that within each series the dimension of the orbit is a linear function of the natural parameter a=1,2,4,8, respectively for f_4,e_6,e_7,e_8. We also obtain explicit representatives in a uniform manner. We observe similar regularities for the centralizers of nilpotent elements in a series and graded components in the associated grading of the ambient Lie algebra. More strikingly, for a greater than one, the degrees of the unipotent characters of the corresponding Chevalley groups, associated to these series through the Springer correspondance are given by polynomials which have uniform expressions in terms of a.Comment: 20 pages, revised version with more formulas for unipotent character

Noncrossing partitions, clusters and the Coxeter plane

When W is a finite Coxeter group of classical type (A, B, or D), noncrossing partitions associated to W and compatibility of almost positive roots in the associated root system are known to be modeled by certain planar diagrams. We show how the classical-type constructions of planar diagrams arise uniformly from projections of small W-orbits to the Coxeter plane. When the construction is applied beyond the classical cases, simple criteria are apparent for noncrossing and for compatibility for W of types H_3 and I_2(m) and less simple criteria can be found for compatibility in types E_6, F_4 and H_4. Our construction also explains why simple combinatorial models are elusive in the larger exceptional types.Comment: Very minor changes, as suggested by the referee. This is essentially the final version, which will appear in Sem. Lothar. Combin. 32 pages. About 12 of the pages are taken up by 29 figure

Nilpotent pairs, dual pairs, and sheets

Recently, V.Ginzburg introduced the notion of a principal nilpotent pair (= pn-pair) in a semisimple Lie algebra {\frak g}. It is a double counterpart of the notion of a regular nilpotent element. A pair (e_1,e_2) of commuting nilpotent elements is called a pn-pair, if the dimension of their simultaneous centralizer is equal to the rank of {\frak g} and some bi-homogeneity condition is satisfied. Ginzburg proved that many familiar results of the `ordinary' theory have analogues for pn-pairs. The aim of this article is to develop the theory of nilpotent pairs a bit further and to present some applications of it to dual pairs and sheets. It is shown that a large portion of Ginzburg's theory can be extended to the pairs whose simultaneous centraliser is of dimension rk{\frak g}+1. Such pairs are called almost pn-pairs. It is worth noting that the very existence of almost pn-pairs is a purely "double" phenomenon, because the dimension of "ordinary" orbits is always even. We prove that to any principal or almost nilpotent pair one naturally associates a dual pair. Moreover, this dual pair is reductive if and only if e_1 and e_2 can be included in commuting sl_2-triples. We also study sheets containing members of pn-pairs. Some cases are described, where these sheets are smooth and admit a section.Comment: 27 pages, LaTeX 2.0

Strictly transversal slices to conjugacy classes in algebraic groups

We show that for every conjugacy class O in a connected semisimple algebraic group G over a field of characteristic good for G one can find a special transversal slice S to the set of conjugacy classes in G such that O intersects S and dim O = codim S.Comment: 38 pages; minor modification

Configurations of lines and models of Lie algebras

The automorphism groups of the 27 lines on the smooth cubic surface or the 28 bitangents to the general quartic plane curve are well-known to be closely related to the Weyl groups of $E\_6$ and $E\_7$. We show how classical subconfigurations of lines, such as double-sixes, triple systems or Steiner sets, are easily constructed from certain models of the exceptional Lie algebras. For ${\mathfrak e}\_7$ and ${\mathfrak e}\_8$ we are lead to beautiful models graded over the octonions, which display these algebras as plane projective geometries of subalgebras. We also interpret the group of the bitangents as a group of transformations of the triangles in the Fano plane, and show how this allows to realize the isomorphism $PSL(3,F\_2)\simeq PSL(2,F\_7)$ in terms of harmonic cubes.Comment: 31 page