221 research outputs found
Combinatorial proof for a stability property of plethysm coefficients
Plethysm coefficients are important structural constants in the representation the-
ory of the symmetric groups and general linear groups. Remarkably, some sequences
of plethysm coefficients stabilize (they are ultimately constants). In this paper we
give a new proof of such a stability property, proved by Brion with geometric representation theory techniques. Our new proof is purely combinatorial: we decompose
plethysm coefficients as a alternating sum of terms counting integer points in poly-
topes, and exhibit bijections between these sets of integer points.Ministerio de Ciencia e Innovación MTM2010–19336Junta de Andalucía FQM–333Junta de Andalucía P12–FQM–269
Plethysm and lattice point counting
We apply lattice point counting methods to compute the multiplicities in the
plethysm of . Our approach gives insight into the asymptotic growth of
the plethysm and makes the problem amenable to computer algebra. We prove an
old conjecture of Howe on the leading term of plethysm. For any partition
of 3,4, or 5 we obtain an explicit formula in and for the
multiplicity of in .Comment: 25 pages including appendix, 1 figure, computational results and code
available at http://thomas-kahle.de/plethysm.html, v2: various improvements,
v3: final version appeared in JFoC
On a curious variant of the -module
We introduce a variant of the much-studied representation of the
symmetric group , which we denote by Our variant gives rise
to a decomposition of the regular representation as a sum of {exterior} powers
of modules This is in contrast to the theorems of
Poincar\'e-Birkhoff-Witt and Thrall which decompose the regular representation
into a sum of symmetrised modules. We show that nearly every known
property of has a counterpart for the module suggesting
connections to the cohomology of configuration spaces via the character
formulas of Sundaram and Welker, to the Eulerian idempotents of Gerstenhaber
and Schack, and to the Hodge decomposition of the complex of injective words
arising from Hochschild homology, due to Hanlon and Hersh.Comment: 26 pages, 2 tables. To appear in Algebraic Combinatorics. Parts of
this paper are included in arXiv:1803.0936
On p-form theories with gauge invariant second order field equations
We explore field theories of a single p-form with equations of motions of
order strictly equal to two and gauge invariance. We give a general method for
the classification of such theories which are extensions to the p-forms of the
Galileon models for scalars. Our classification scheme allows to compute an
upper bound on the number of different such theories depending on p and on the
space-time dimension. We are also able to build a non trivial Galileon like
theory for a 3-form with gauge invariance and an action which is polynomial
into the derivatives of the form. This theory has gauge invariant field
equations but an action which is not, like a Chern-Simons theory. Hence the
recently discovered no-go theorem stating that there are no non trivial gauge
invariant vector Galileons (which we are also able here to confirm with our
method) does not extend to other odd p cases.Comment: 29 page
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