表征理论/光谱理论/K理论与同源学术速递[1.10]
Update！H5支持摘要折叠，体验更佳！点击阅读原文访问arxivdaily.com，涵盖CS|物理|数学|经济|统计|金融|生物|电气领域，更有搜索、收藏等功能！
math.RT表征理论，共计4篇
math.SP光谱理论，共计1篇
math.KTK理论与同源，共计0篇
1.math.RT表征理论:
【1】 Summation formulae for quadrics
标题：二次曲面的求和公式
链接：https://arxiv.org/abs/2201.02583
摘要：We prove a Poisson summation formula for the zero locus of a quadratic form
in an even number of variables with no assumption on the support of the
functions involved. The key novelty in the formula is that all "boundary terms"
are given either by constants or sums over smaller quadrics related to the
original quadric. We also discuss the link with the classical problem of
estimating the number of solutions of a quadratic form in an even number of
variables. To prove the summation formula we compute (the Arthur truncated)
theta lift of the trivial representation of $\mathrm{SL}_2(\mathbb{A}_F)$. As
previously observed by Ginzburg, Rallis, and Soudry, this is an analogue for
orthogonal groups on vector spaces of even dimension of the global
Schr\"odinger representation of the metaplectic group.
【2】 Graded irreducible representations of Leavitt path algebras: a new type and complete classification
标题：Leavitt路代数的分次不可约表示：一种新的完全分类
链接：https://arxiv.org/abs/2201.02446
摘要：We present a new class of graded irreducible representations of a Leavitt
path algebra. This class is new in the sense that its representation space is
not isomorphic to any of the existing simple Chen modules. The corresponding
graded simple modules complete the list of Chen modules which are graded,
creating an exhaustive class: the annihilator of any graded simple module is
equal to the annihilator of either a graded Chen module or a module of this new
type.
Our characterization of graded primitive ideals of a Leavitt path algebra in
terms of the properties of the underlying graph is the main tool for proving
the completeness of such classification. We also point out a problem with the
characterization of primitive ideals of a Leavitt path algebra in [K. M.
Rangaswamy, Theory of prime ideals of Leavitt path algebras over arbitrary
graphs, J. Algebra 375 (2013), 73 -- 90].
【3】 Bases for infinite dimensional simple $\mathfrak{osp}(1|2n)$-modules respecting the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$
链接：https://arxiv.org/abs/2201.02438
备注：31 pages
摘要：We study the effects of the branching $\mathfrak{osp}(1|2n)\supset
\mathfrak{gl}(n)$ on a particular class of simple infinite-dimensional
$\mathfrak{osp}(1|2n)$-modules $L(p)$ characterized by a positive integer $p$.
In the first part we use combinatorial methods such as Young tableaux and Young
subgroups to construct a new basis for $L(p)$ that respects this branching and
we express the basis elements explicitly in two distinct ways. First as
monomials of negative root vectors of $\mathfrak{gl}(n)$ acting on the
$\mathfrak{gl}(n)$-highest weight vectors in $L(p)$ and then as polynomials in
the generators of $\mathfrak{osp}(1|2n)$ acting on the
$\mathfrak{osp}(1|2n)$-lowest weight vector in $L(p)$. In the second part we
use extremal projectors and the theory of Mickelsson-Zhelobenko algebras to
give new explicit constructions of raising and lowering operators related to
the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$. We use the
raising operators to give new expressions for the elements of the
Gel'fand-Zetlin basis for $L(p)$ as monomials of operators from
$U(\mathfrak{osp}(1|2n))$ acting on the $\mathfrak{osp}(1|2n)$-lowest weight
vector in $L(p)$. We observe that the Gel'fand-Zetlin basis for $L(p)$ is
related to the basis constructed earlier in the paper by a triangular
transition matrix. We end the paper with a detailed example treating the case
$n=3$.
【4】 Perfect bases in representation theory: three mountains and their springs
标题：表征理论的完美基础：三座山及其泉水
链接：https://arxiv.org/abs/2201.02289
备注：18 pages, ICM 2022 proceedings
摘要：In order to give a combinatorial descriptions of tensor product multiplicites
for semisimple groups, it is useful to find bases for representations which are
compatible with the actions of Chevalley generators of the Lie algebra. There
are three known examples of such bases, each of which flows from geometric or
algebraic mountain. Remarkably, each mountain gives the same combinatorial
shadow: the crystal B(infty) and the Mirkovic-Vilonen polytopes. In order to
distinguish between the three bases, we introduce measures supported on these
polytopes. We also report on the interaction of these bases with the cluster
structure on the coordinate ring of the maximal unipotent subgroup.
2.math.SP光谱理论:
【1】 Hardy inequalities for magnetic $p$-Laplacians
标题：磁性$p$-Laplacian的Hardy不等式
链接：https://arxiv.org/abs/2201.02482
备注：20 pages
摘要：Improved Hardy inequalities for the $p$-Laplacian due to adding magnetic
fields are established, while other expected results are stated as conjectures.
Some general $L^p$ magnetic-free Hardy inequalities in the spirit of Allegretto
and Huang [Nonlinear Anal. 32 (1998)] are also considered.
3.math.KTK理论与同源:
机器翻译，仅供参考
点击“阅读原文”获取带摘要的学术速递