27 research outputs found
Ordered set partitions and the -Hecke algebra
Let the symmetric group act on the polynomial ring
by variable
permutation. The coinvariant algebra is the graded -module , where is the ideal in
generated by invariant polynomials with vanishing
constant term. Haglund, Rhoades, and Shimozono introduced a new quotient
of the polynomial ring depending on two
positive integers which reduces to the classical coinvariant algebra
of the symmetric group when . The quotient
carries the structure of a graded -module; Haglund et. al.
determine its graded isomorphism type and relate it to the Delta Conjecture in
the theory of Macdonald polynomials. We introduce and study a related quotient
of which carries a graded action of the
0-Hecke algebra , where is an arbitrary field. We prove
0-Hecke analogs of the results of Haglund, Rhoades, and Shimozono. In the
classical case , we recover earlier results of Huang concerning the
0-Hecke action on the coinvariant algebra.Comment: 30 pages. Corrected the bijection following Definition 3.
The invariant rings of the Sylow groups of GU(3,q2), GU(4,q2), Sp(4,q) and O+(4,q) in the natural characteristic
Let G be a Sylow p-subgroup of the unitary groups GU(3, q2),
GU(4, q2), the symplectic group Sp(4, q) and, for q odd, the
orthogonal group O +(4, q). In this paper we construct a presenta tion for the invariant ring of G acting on the natural module.
In particular we prove that these rings are generated by orbit
products of variables and certain invariant polynomials which
are images under Steenrod operations, applied to the respective
invariant form defining the corresponding classical group. We also
show that these generators form a SAGBI basis and the invariant
ring for G is a complete intersection.info:eu-repo/semantics/publishedVersio
The nullcone in the multi-vector representation of the symplectic group and related combinatorics
We study the nullcone in the multi-vector representation of the symplectic
group with respect to a joint action of the general linear group and the
symplectic group. By extracting an algebra over a distributive lattice
structure from the coordinate ring of the nullcone, we describe a toric
degeneration and standard monomial theory of the nullcone in terms of double
tableaux and integral points in a convex polyhedral cone.Comment: 21 pages, v2: title changed, typos and errors correcte
On Harmonic Elements for Semi-simple Lie Algebras
AbstractLet g be a semi-simple complex Lie algebra and g=n−⊕h⊕n its triangular decomposition. Let U(g), resp. Uq(g), be its enveloping algebra, resp. its quantized enveloping algebra. This article gives a quantum approach to the combinatorics of (classical) harmonic elements and Kostant's generalized exponents for g. A quantum analogue of the space of harmonic elements has been given by A. Joseph and G. Letzter (1994, Amer. J. Math.116, 127–177). On the one hand, we give specialization results concerning harmonic elements, central elements of Uq(g), and the decomposition of Joseph and Letzter (cited above). For g=sln+1, we describe the specialization of quantum harmonic space in the N-filtered algebra U(sln+1) as the materialization of a theorem of A. Lascoux et al. (1995, Lett. Math. Phys.35, 359–374). This enables us to study a Joseph–Letzter decomposition in the algebra U(sln+1). On the other hand, we prove that highest weight harmonic elements can be calculated in terms of the dual of Lusztig's canonical basis. In the simply laced case, we parametrize a basis of n-invariants of minimal primitive quotients by the set C0 of integral points of a convex cone
Low-Rank Parity-Check Codes over Galois Rings
Low-rank parity-check (LRPC) are rank-metric codes over finite fields, which
have been proposed by Gaborit et al. (2013) for cryptographic applications.
Inspired by a recent adaption of Gabidulin codes to certain finite rings by
Kamche et al. (2019), we define and study LRPC codes over Galois rings - a wide
class of finite commutative rings. We give a decoding algorithm similar to
Gaborit et al.'s decoder, based on simple linear-algebraic operations. We
derive an upper bound on the failure probability of the decoder, which is
significantly more involved than in the case of finite fields. The bound
depends only on the rank of an error, i.e., is independent of its free rank.
Further, we analyze the complexity of the decoder. We obtain that there is a
class of LRPC codes over a Galois ring that can decode roughly the same number
of errors as a Gabidulin code with the same code parameters, but faster than
the currently best decoder for Gabidulin codes. However, the price that one
needs to pay is a small failure probability, which we can bound from above.Comment: 37 pages, 1 figure, extended version of arXiv:2001.0480