scholarly journals Representation stability for sequences of 0-Hecke modules

2021 ◽  
Vol 4 (4) ◽  
pp. 619-661
Author(s):  
Robert P. Laudone
Keyword(s):  
Representation Stability ◽  
Hecke Modules

scholarly journals Hecke modules for arithmetic groups via bivariant K-theory

2018 ◽  
Vol 3 (4) ◽  
pp. 631-656 ◽  
Author(s):  
Bram Mesland ◽  
Mehmet Haluk Şengün
Keyword(s):  
Arithmetic Groups ◽  
K Theory ◽  
Hecke Modules

Detecting Large Simple Rational Hecke Modules for Γ0(N) via Congruences

2018 ◽  
Vol 2020 (19) ◽  
pp. 6149-6168
Author(s):  
Michael Lipnowski ◽  
George J Schaeffer
Keyword(s):  
Modular Forms ◽  
Class Number ◽  
Target Space ◽  
Large Set ◽  
Hecke Eigenvalues ◽  
Artin's Conjecture ◽  
Number Problem ◽  
Novel Method ◽  
Hecke Modules ◽  
Primitive Roots

Abstract We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_{2}(\Gamma _{0}(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_{0}(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. We prove conditional bounds, the strongest of which is $d\gg _{\epsilon } N^{1/2-\epsilon }$ over a large set of primes $N$, contingent on Soundararajan’s heuristics for the class number problem and Artin’s conjecture on primitive roots. For prime levels $N\equiv 7\mod 8,$ our method yields an unconditional bound of $d\geq \log _{2}\log _{2}(\frac{N}{8})$, which is larger than the known bound of $d\gg \sqrt{\log \log N}$ due to Murty–Sinha and Royer. A stronger unconditional bound of $d\gg \log N$ can be obtained in more specialized (but infinitely many) cases. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of $S_{k}(\textrm{SL}_{2}(\mathbb{Z});\mathbb{Q})$.

scholarly journals Stability in the high-dimensional cohomology of congruence subgroups

2020 ◽  
Vol 156 (4) ◽  
pp. 822-861
Author(s):  
Jeremy Miller ◽  
Rohit Nagpal ◽  
Peter Patzt
Keyword(s):  
Congruence Subgroup ◽  
Stability Result ◽  
High Dimensional ◽  
Vanishing Theorem ◽  
Congruence Subgroups ◽  
Codimension One ◽  
Representation Stability ◽  
Steinberg Module ◽  
Finiteness Properties ◽  
Special Case

We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

scholarly journals Representation Stability of Power Sets and Square Free Polynomials

2015 ◽  
Vol 67 (5) ◽  
pp. 1024-1045
Author(s):  
Samia Ashraf ◽  
Haniya Azam ◽  
Barbu Berceanu
Keyword(s):  
Symmetric Group ◽  
Braid Group ◽  
Point Of View ◽  
Pure Braid Group ◽  
Representation Stability ◽  
Stability Point ◽  
Power Set ◽  
The Stability

AbstractThe symmetric group 𝓢n acts on the power set 𝓟(n) and also on the set of square free polynomials in n variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

scholarly journals Homological Stability for Spaces of Commuting Elements in Lie Groups

2020 ◽  
Author(s):  
Daniel A Ramras ◽  
Mentor Stafa
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Stable Range ◽  
Rational Homology ◽  
Infinite Dimensional ◽  
Fixed Group ◽  
Character Varieties ◽  
Representation Stability ◽  
Equivariant Homology ◽  
Homological Stability

Abstract In this paper, we study homological stability for spaces $\textrm{Hom}({{\mathbb{Z}}}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, $\textrm{Comm}(G)$ and $B_{\textrm{com}} G$, introduced by Cohen–Stafa and Adem–Cohen–Torres-Giese, respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability—in particular, the theory of $\textrm{FI}_W$-modules developed by Church–Ellenberg–Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.

scholarly journals Modular forms of half-integral weights on SL(2, ℤ)

2014 ◽  
Vol 215 ◽  
pp. 1-66
Author(s):  
Yifan Yang
Keyword(s):  
Modular Forms ◽  
Analogous Result ◽  
Quadratic Character ◽  
Hecke Modules ◽  
Shimura Correspondence

AbstractIn this paper, we prove that, for an integerrwith (r, 6) = 1 and 0< r <24 and a nonnegative even integers, the setis isomorphic toas Hecke modules under the Shimura correspondence. HereMs(1) denotes the space of modular forms of weightis the space of newforms of weight 2kon Γ0(6) that are eigenfunctions with eigenvalues€2and€3for Atkin-Lehner involutionsW2andW3, respectively, and the notation ⊕(12/.) means the twist by the quadratic character (12/-). There is also an analogous result for the cases (r, 6) = 3.

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