Relevant bibliographies by topics / Group-Weights conjecture

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Academic literature on the topic 'Group-Weights conjecture'

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Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Group-Weights conjecture"

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Puig, Lluis. "Parameterization of Irreducible Characters for p-Solvable Groups." Algebra Colloquium 19, no. 01 (March 2012): 1–40. http://dx.doi.org/10.1142/s1005386712000028.

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The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulated a refinement of Alperin's conjecture involving ordinary irreducible characters — with their defect — and, in 2000, Geoffrey Robinson proved that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in a previous paper can be suitably refined to provide, up to the choice of a polarization ω, a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of absolutely irreducible characters of G and of G-conjugacy classes of suitable inductive weights, preserving blocks and defects.
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Puig, Lluis. "Weight Parameterization of Simple Modules for p-Solvable Groups." Algebra Colloquium 20, no. 01 (January 16, 2013): 1–46. http://dx.doi.org/10.1142/s1005386713000023.

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The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture affirming that the number of G-conjugacy classes of weights of G coincides with the number of isomorphism classes of simple kG-modules, where k is an algebraically closed field of characteristic p. Thirty years ago, Tetsuro Okuyama already proved that in the class of p-solvable groups this conjecture holds. In this paper, for the p-solvable groups, on the one hand we exhibit a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of isomorphism classes of simple kG-modules M and of G-conjugacy classes of weights (R,Y), up to the choice of a polarization. On the other hand, we determine the relationship between a multiplicity module of M and Y. In an Appendix, we show that the bijection defined by Gabriel Navarro for the groups of odd order coincides with our bijection for a particular choice of the polarization.
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Li, Conghui, and Zhenye Li. "The Inductive Blockwise Alperin Weight Condition for PSp4(q)." Algebra Colloquium 26, no. 03 (August 12, 2019): 361–86. http://dx.doi.org/10.1142/s1005386719000270.

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Let G be a finite group and ℓ be any prime dividing [Formula: see text]. The blockwise Alperin weight conjecture states that the number of the irreducible Brauer characters in an ℓ-block B of G equals the number of the G-conjugacy classes of ℓ-weights belonging to B. Recently, this conjecture has been reduced to the simple groups, which means that to prove the blockwise Alperin weight conjecture, it suffices to prove that all non-abelian simple groups satisfy the inductive blockwise Alperin weight condition. In this paper, we verify this inductive condition for the finite simple groups [Formula: see text] and non-defining characteristic, where q is a power of an odd prime.
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Feng, Zhicheng, Conghui Li, and Zhenye Li. "The Inductive Blockwise Alperin Weight Condition for PSL(3, q)." Algebra Colloquium 24, no. 01 (February 15, 2017): 123–52. http://dx.doi.org/10.1142/s1005386717000086.

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The blockwise Alperin weight conjecture assets that for any finite group G and any prime l, the number of the Brauer characters in an l-block B equals the number of the G-conjugacy classes of l-weights belonging to B. Recently, the inductive blockwise Alperin weight condition has been introduced such that the blockwise Alperin weight conjecture holds if all non-abelian simple groups satisfy these conditions. We will verify the inductive blockwise Alperin weight condition for the finite simple groups PSL(3, q) in this paper.
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Abe, Noriyuki, and Masaharu Kaneda. "THE LOEWY STRUCTURE OF -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS." Journal of the Institute of Mathematics of Jussieu 16, no. 4 (October 2, 2015): 887–98. http://dx.doi.org/10.1017/s1474748015000274.

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Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.
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GUILHOT, JÉRÉMIE, and CÉDRIC LECOUVEY. "ISOMORPHIC INDUCED MODULES AND DYNKIN DIAGRAM AUTOMORPHISMS OF SEMISIMPLE LIE ALGEBRAS." Glasgow Mathematical Journal 58, no. 1 (July 21, 2015): 187–203. http://dx.doi.org/10.1017/s0017089515000142.

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AbstractConsider a simple Lie algebra $\mathfrak{g}$ and $\overline{\mathfrak{g}}$ ⊂ $\mathfrak{g}$ a Levi subalgebra. Two irreducible $\overline{\mathfrak{g}}$-modules yield isomorphic inductions to $\mathfrak{g}$ when their highest weights coincide up to conjugation by an element of the Weyl group W of $\mathfrak{g}$ which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$. In this paper, we study the converse problem: given two irreducible $\overline{\mathfrak{g}}$-modules of highest weight μ and ν whose inductions to $\mathfrak{g}$ are isomorphic, can we conclude that μ and ν are conjugate under the action of an element of W which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$? We conjecture this is true in general. We prove this conjecture in type A and, for the other root systems, in various situations providing μ and ν satisfy additional hypotheses. Our result can be interpreted as an analogue for branching coefficients of the main result of Rajan [6] on tensor product multiplicities.
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Richards, Matthew J. "Some decomposition numbers for Hecke algebras of general linear groups." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 3 (April 1996): 383–402. http://dx.doi.org/10.1017/s0305004100074296.

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The theorem which is still known as Nakayama's Conjecture shows how the modular characters of the symmetric group Sn can be divided into blocks of various weights, those with the same weight having similar properties. In fact, all blocks of weight one have essentially the same decomposition numbers and these are easy to describe. In recent work, Scopes [16, 17] has shown that there are essentially only finitely many possibilities for the decomposition numbers for blocks of any given weight, and has given a bound for the number. We develop the combinatorics implicit in her work, and so establish an improved bound.
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Feigin, Evgeny, Michael Finkelberg, and Peter Littelmann. "Symplectic Degenerate Flag Varieties." Canadian Journal of Mathematics 66, no. 6 (December 1, 2014): 1250–86. http://dx.doi.org/10.4153/cjm-2013-038-6.

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AbstractA simple finite dimensional module Vλ of a simple complex algebraic group G is naturally endowed with a filtration induced by the PBW-filtration of U(Lie G). The associated graded space is a module for the group Ga, which can be roughly described as a semi-direct product of a Borel subgroup of G and a large commutative unipotent group . In analogy to the flag variety ℱλ = G:[vλ] ⊂ ℙ(Vλ), we call the closure of the Ga-orbit through the highest weight line the degenerate flag variety . In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of G being the symplectic group. The symplectic case is important for the conjecture because it is the first known case where, even for fundamental weights ω, the varieties differ from Fω. We give an explicit construction of the varieties and construct desingularizations, similar to the Bott–Samelson resolutions in the classical case. We prove that are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel–Weil theorem and obtain a q-character formula for the characters of irreducible Sp2n-modules via the Atiyah–Bott–Lefschetz fixed points formula.
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Michler, G. O., and J. B. Olsson. "Weights for Covering Groups of Symmetric and Alternating Groups, р ≠ 2." Canadian Journal of Mathematics 43, no. 4 (August 1, 1991): 792–813. http://dx.doi.org/10.4153/cjm-1991-045-6.

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In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.
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10

Noriyuki, Abe, and Kaneda Masaharu. "Loewy series of parabolically induced -Verma modules." Journal of the Institute of Mathematics of Jussieu 14, no. 1 (March 28, 2014): 185–220. http://dx.doi.org/10.1017/s1474748014000012.

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AbstractWe show that the modules for the Frobenius kernel of a reductive algebraic group over an algebraically closed field of positive characteristic $p$ induced from the $p$-regular blocks of its parabolic subgroups can be $\mathbb{Z}$-graded. In particular, we obtain that the modules induced from the simple modules of $p$-regular highest weights are rigid and determine their Loewy series, assuming the Lusztig conjecture on the irreducible characters for the reductive algebraic groups, which is now a theorem for large $p$. We say that a module is rigid if and only if it admits a unique filtration of minimal length with each subquotient semisimple, in which case the filtration is called the Loewy series.
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Dissertations / Theses on the topic "Group-Weights conjecture"

1

Tay, Julian Boon Kai. "Poincaré Polynomial of FJRW Rings and the Group-Weights Conjecture." BYU ScholarsArchive, 2013. https://scholarsarchive.byu.edu/etd/3604.

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FJRW-theory is a recent advancement in singularity theory arising from physics. The FJRW-theory is a graded vector space constructed from a quasihomogeneous weighted polynomial and symmetry group, but it has been conjectured that the theory only depends on the weights of the polynomial and the group. In this thesis, I prove this conjecture using Poincaré polynomials and Koszul complexes. By constructing the Koszul complex of the state space, we have found an expression for the Poincaré polynomial of the state space for a given polynomial and associated group. This Poincaré polynomial is defined over the representation ring of a group in order for us to take G-invariants. It turns out that the construction of the Koszul complex is independent of the choice of polynomial, which proves our conjecture that two different polynomials with the same weights will have isomorphic FJRW rings as long as the associated groups are the same.
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