27 research outputs found

    Ordered set partitions and the 00-Hecke algebra

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    Let the symmetric group Sn\mathfrak{S}_n act on the polynomial ring Q[xn]=Q[x1,…,xn]\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n] by variable permutation. The coinvariant algebra is the graded Sn\mathfrak{S}_n-module Rn:=Q[xn]/InR_n := {\mathbb{Q}[\mathbf{x}_n]} / {I_n}, where InI_n is the ideal in Q[xn]\mathbb{Q}[\mathbf{x}_n] generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient Rn,kR_{n,k} of the polynomial ring Q[xn]\mathbb{Q}[\mathbf{x}_n] depending on two positive integers k≤nk \leq n which reduces to the classical coinvariant algebra of the symmetric group Sn\mathfrak{S}_n when k=nk = n. The quotient Rn,kR_{n,k} carries the structure of a graded Sn\mathfrak{S}_n-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 Sn,kS_{n,k} of F[xn]\mathbb{F}[\mathbf{x}_n] which carries a graded action of the 0-Hecke algebra Hn(0)H_n(0), where F\mathbb{F} is an arbitrary field. We prove 0-Hecke analogs of the results of Haglund, Rhoades, and Shimozono. In the classical case k=nk = n, 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

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    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

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    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

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    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

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    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
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