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Journal articles on the topic 'Group-Weights conjecture'
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1
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.
3
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.
4
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.
5
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.
6
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.
7
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.
9
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.
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.
11
Herzig, Florian, Daniel Le, and Stefano Morra. "On mod local-global compatibility for in the ordinary case." Compositio Mathematica 153, no. 11 (August 25, 2017): 2215–86. http://dx.doi.org/10.1112/s0010437x17007357.
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Suppose that $F/F^{+}$ is a CM extension of number fields in which the prime $p$ splits completely and every other prime is unramified. Fix a place $w|p$ of $F$. Suppose that $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$ is a continuous irreducible Galois representation such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. If $\overline{r}$ is automorphic, and some suitable technical conditions hold, we show that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ can be recovered from the $\text{GL}_{3}(F_{w})$-action on a space of mod $p$ automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for $\overline{r}$, show the existence of an ordinary lifting of $\overline{r}$, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations $\overline{r}$ to which our main theorem applies.
12
Ceulemans, A., E. Lijnen, P. W. Fowler, R. B. Mallion, and T. Pisanski. "Graph theory and the Jahn–Teller theorem." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2140 (November 30, 2011): 971–89. http://dx.doi.org/10.1098/rspa.2011.0508.
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The Jahn–Teller (JT) theorem predicts spontaneous symmetry breaking and lifting of degeneracy in degenerate electronic states of (nonlinear) molecular and solid-state systems. In these cases, degeneracy is lifted by geometric distortion. Molecular problems are often modelled using spectral theory for weighted graphs, and the present paper turns this process around and reformulates the JT theorem for general vertex- and edge-weighted graphs themselves. If the eigenvectors and eigenvalues of a general graph are considered as orbitals and energy levels (respectively) to be occupied by electrons, then degeneracy of states can be resolved by a non-totally symmetric re-weighting of edges and, where necessary, vertices. This leads to the conjecture that whenever the spectrum of a graph contains a set of bonding or anti-bonding degenerate eigenvalues, the roots of the Hamiltonian matrix over this set will show a linear dependence on edge distortions, which has the effect of lifting the degeneracy. When the degenerate level is non-bonding, distortions of vertex weights have to be included to obtain a full resolution of the eigenspace of the degeneracy. Explicit treatments are given for examples of the octahedral graph, where the degeneracy to be lifted is forced by symmetry, and the phenalenyl graph, where the degeneracy is accidental in terms of the automorphism group.
13
Jorgenson, Jay, Lejla Smajlović, and Holger Then. "Certain aspects of holomorphic function theory on some genus-zero arithmetic groups." LMS Journal of Computation and Mathematics 19, no. 2 (2016): 360–81. http://dx.doi.org/10.1112/s1461157016000425.
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There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group $\operatorname{PSL}(2,\mathbb{Z})$, including the following statements: the ring of holomorphic modular forms is generated by the holomorphic Eisenstein series of weights four and six, denoted by $E_{4}$ and $E_{6}$; the smallest-weight cusp form $\unicode[STIX]{x1D6E5}$ has weight twelve and can be written as a polynomial in $E_{4}$ and $E_{6}$; and the Hauptmodul $j$ can be written as a multiple of $E_{4}^{3}$ divided by $\unicode[STIX]{x1D6E5}$. The goal of the present article is to seek generalizations of these results to some other genus-zero arithmetic groups $\unicode[STIX]{x1D6E4}_{0}(N)^{+}$ with square-free level $N$, which are related to ‘Monstrous moonshine conjectures’. Certain aspects of our results are generated from extensive computer analysis; as a result, many of the space-consuming results are made available on a publicly accessible web site. However, we do present in this article specific results for certain low-level groups.
14
LE, DANIEL, BAO V. LE HUNG, BRANDON LEVIN, and STEFANO MORRA. "SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE." Forum of Mathematics, Pi 8 (2020). http://dx.doi.org/10.1017/fmp.2020.1.
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We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $p$ . This is a generalization to $\text{GL}_{3}$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $n$ -dimensional Galois representations’, Duke Math. J. 149(1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math. 212(1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $\text{GL}_{3}(\mathbb{F}_{q})$ .
15
Navarro, Gabriel, and Benjamin Sambale. "Weights and Nilpotent Subgroups." International Mathematics Research Notices, November 27, 2019. http://dx.doi.org/10.1093/imrn/rnz195.
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Abstract In a finite group $G$, we consider nilpotent weights and prove a $\pi $-version of the Alperin Weight Conjecture for certain $\pi $-separable groups. This widely generalizes an earlier result by I. M. Isaacs and the 1st author.
16
Howe, Sean. "Overconvergent modular forms are highest-weight vectors in the Hodge-Tate weight zero part of completed cohomology." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.16.
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Abstract We construct a $(\mathfrak {gl}_2, B(\mathbb {Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $0$ of a sheaf on $\mathbb {P}^1$ , landing in the compactly supported completed $\mathbb {C}_p$ -cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest-weight vector for any overconvergent modular form of infinitesimal weight not equal to $1$ . For classical weight $k\geq 2$ , the Verma has an algebraic quotient $H^1(\mathbb {P}^1, \mathcal {O}(-k))$ , and on classical forms, the pairing factors through this quotient, giving a geometric description of ‘half’ of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $\mathbb {P}^1$ . Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan, but the perspective here is different, and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology.
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DARMON, HENRI, ALAN LAUDER, and VICTOR ROTGER. "STARK POINTS AND -ADIC ITERATED INTEGRALS ATTACHED TO MODULAR FORMS OF WEIGHT ONE." Forum of Mathematics, Pi 3 (2015). http://dx.doi.org/10.1017/fmp.2015.7.
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Let$E$be an elliptic curve over$\mathbb{Q}$, and let${\it\varrho}_{\flat }$and${\it\varrho}_{\sharp }$be odd two-dimensional Artin representations for which${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms$f$,$g$, and$h$of respective weights two, one, and one, giving rise to$E$,${\it\varrho}_{\flat }$, and${\it\varrho}_{\sharp }$via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain$p$-adic iterated integralsattached to the triple$(f,g,h)$, which are$p$-adic avatars of the leading term of the Hasse–Weil–Artin$L$-series$L(E,{\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp },s)$when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on$E$—referred to asStark points—which are defined over the number field cut out by${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$. This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weight-one forms. It is proved when$g$and$h$are binary theta series attached to a common imaginary quadratic field in which$p$splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing$p$-adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintani-type cycles on${\mathcal{H}}_{p}\times {\mathcal{H}}$), and extensions of$\mathbb{Q}$with Galois group a central extension of the dihedral group$D_{2n}$or of one of the exceptional subgroups$A_{4}$,$S_{4}$, and$A_{5}$of$\mathbf{PGL}_{2}(\mathbb{C})$.
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Bagchi, Arjun, Poulami Nandi, Amartya Saha, and Zodinmawia. "BMS modular diaries: torus one-point function." Journal of High Energy Physics 2020, no. 11 (November 2020). http://dx.doi.org/10.1007/jhep11(2020)065.
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Abstract Two dimensional field theories invariant under the Bondi-Metzner-Sachs (BMS) group are conjectured to be dual to asymptotically flat spacetimes in three dimensions. In this paper, we continue our investigations of the modular properties of these field theories. In particular, we focus on the BMS torus one-point function. We use two different methods to arrive at expressions for asymptotic structure constants for general states in the theory utilising modular properties of the torus one-point function. We then concentrate on the BMS highest weight representation, and derive a host of new results, the most important of which is the BMS torus block. In a particular limit of large weights, we derive the leading and sub-leading pieces of the BMS torus block, which we then use to rederive an expression for the asymptotic structure constants for BMS primaries. Finally, we perform a bulk computation of a probe scalar in the background of a flatspace cosmological solution based on the geodesic approximation to reproduce our field theoretic results.