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1. Character bounds for regular semisimple elements and asymptotic results on Thompson’s conjecture

   https://doi.org/10.1007/s00209-022-03193-3  

   Larsen, Michael; Taylor, Jay; Tiep, Pham Huu (February 2023, Mathematische Zeitschrift)

   Abstract For every integer k there exists a bound $$B=B(k)$$ B = B ( k ) such that if the characteristic polynomial of $$g\in \textrm{SL}_n(q)$$ g ∈ SL n ( q ) is the product of $$\le k$$ ≤ k pairwise distinct monic irreducible polynomials over $$\mathbb {F}_q$$ F q , then every element x of $$\textrm{SL}_n(q)$$ SL n ( q ) of support at least B is the product of two conjugates of g . We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions ( p ,  q ), in the special case that $$n=p$$ n = p is prime, if g has order $$\frac{q^p-1}{q-1}$$ q p - 1 q - 1 , then every non-scalar element $$x \in \textrm{SL}_p(q)$$ x ∈ SL p ( q ) is the product of two conjugates of g . The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups. 

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2. Set Partitions, Fermions, and Skein Relations

   https://doi.org/10.1093/imrn/rnac110  

   Kim, Jesse; Rhoades, Brendon (May 2022, International Mathematics Research Notices)

   Abstract Let $$\Theta _n = (\theta _1, \dots , \theta _n)$$ and $$\Xi _n = (\xi _1, \dots , \xi _n)$$ be two lists of $$n$$ variables, and consider the diagonal action of $${{\mathfrak {S}}}_n$$ on the exterior algebra $$\wedge \{ \Theta _n, \Xi _n \}$$ generated by these variables. Jongwon Kim and the 2nd author defined and studied the fermionic diagonal coinvariant ring$$FDR_n$$ obtained from $$\wedge \{ \Theta _n, \Xi _n \}$$ by modding out by the ideal generated by the $${{\mathfrak {S}}}_n$$-invariants with vanishing constant term. On the other hand, the 2nd author described an action of $${{\mathfrak {S}}}_n$$ on the vector space with basis given by noncrossing set partitions of $$\{1,\dots ,n\}$$ using a novel family of skein relations that resolve crossings in set partitions. We give an isomorphism between a natural Catalan-dimensional submodule of $$FDR_n$$ and the skein representation. To do this, we show that set partition skein relations arise naturally in the context of exterior algebras. Our approach yields an $${{\mathfrak {S}}}_n$$-equivariant way to resolve crossings in set partitions. We use fermions to clarify, sharpen, and extend the theory of set partition crossing resolution. 

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3. Characterizing Schwarz maps by tracial inequalities

   https://doi.org/10.1007/s11005-023-01636-4  

   Carlen, Eric; Müller-Hermes, Alexander (February 2023, Letters in Mathematical Physics)

   Abstract Let $$\phi $$ ϕ be a positive map from the $$n\times n$$ n × n matrices $$\mathcal {M}_n$$ M n to the $$m\times m$$ m × m matrices $$\mathcal {M}_m$$ M m . It is known that $$\phi $$ ϕ is 2-positive if and only if for all $$K\in \mathcal {M}_n$$ K ∈ M n and all strictly positive $$X\in \mathcal {M}_n$$ X ∈ M n , $$\phi (K^*X^{-1}K) \geqslant \phi (K)^*\phi (X)^{-1}\phi (K)$$ ϕ ( K ∗ X - 1 K ) ⩾ ϕ ( K ) ∗ ϕ ( X ) - 1 ϕ ( K ) . This inequality is not generally true if $$\phi $$ ϕ is merely a Schwarz map. We show that the corresponding tracial inequality $${{\,\textrm{Tr}\,}}[\phi (K^*X^{-1}K)] \geqslant {{\,\textrm{Tr}\,}}[\phi (K)^*\phi (X)^{-1}\phi (K)]$$ Tr [ ϕ ( K ∗ X - 1 K ) ] ⩾ Tr [ ϕ ( K ) ∗ ϕ ( X ) - 1 ϕ ( K ) ] holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results. 

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4. Morphisms of Character Varieties

   https://doi.org/10.1093/imrn/rnae124  

   Cotner, Sean (June 2024, International Mathematics Research Notices)

   Abstract Let $$k$$ be a field, let $$H \subset G$$ be (possibly disconnected) reductive groups over $$k$$, and let $$\Gamma $$ be a finitely generated group. Vinberg and Martin have shown that the induced morphism $$\underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , H)//H \to \underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , G)//G$$ is finite. In this note, we generalize this result (with a significantly different proof) by replacing $$k$$ with an arbitrary locally Noetherian scheme, answering a question of Dat. Along the way, we use Bruhat–Tits theory to establish a few apparently new results about integral models of reductive groups over discrete valuation rings. 

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5. Evidence for the singly Cabibbo-suppressed decay $$ {\Omega}_c^0\to {\Xi}^{-}{\pi}^{+} $$ and search for $$ {\Omega}_c^0\to {\Xi}^{-}{K}^{+} $$ and Ω−K+ decays at Belle

   https://doi.org/10.1007/JHEP01(2023)055  

   Han, X.; Jia, S.; Yuan, L.; Shen, C. P.; Adachi, I.; Ahn, J. K.; Aihara, H.; Asner, D. M.; Aushev, T.; Ayad, R.; et al (January 2023, Journal of High Energy Physics)

   A bstract Using a data sample of 980 fb − 1 collected with the Belle detector at the KEKB asymmetric-energy e + e − collider, we study for the first time the singly Cabibbo-suppressed decays $$ {\Omega}_c^0\to {\Xi}^{-}{\pi}^{+} $$ Ω c 0 → Ξ − π + and Ω − K + and the doubly Cabibbo-suppressed decay $$ {\Omega}_c^0\to {\Xi}^{-}{K}^{+} $$ Ω c 0 → Ξ − K + . Evidence for an $$ {\Omega}_c^0 $$ Ω c 0 signal in the $$ {\Omega}_c^0 $$ Ω c 0 → Ξ − π + mode is reported with a significance of 4 . 5 σ including systematic uncertainties. The ratio of branching fractions to the normalization mode $$ {\Omega}_c^0 $$ Ω c 0 → Ω − π + is measured to be $$ \mathcal{B}\left({\Omega}_c^0\to {\Xi}^{-}{\pi}^{+}\right)/\mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{\pi}^{+}\right)=0.253\pm 0.052\left(\textrm{stat}.\right)\pm 0.030\left(\textrm{syst}.\right). $$ B Ω c 0 → Ξ − π + / B Ω c 0 → Ω − π + = 0.253 ± 0.052 stat . ± 0.030 syst . . No significant signals of $$ {\Omega}_c^0\to {\Xi}^{-}{K}^{+} $$ Ω c 0 → Ξ − K + and Ω − K + modes are found. The upper limits at 90% confidence level on ratios of branching fractions are determined to be $$ \mathcal{B}\left({\Omega}_c^0\to {\Xi}^{-}{K}^{+}\right)/\mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{\pi}^{+}\right)<0.070 $$ B Ω c 0 → Ξ − K + / B Ω c 0 → Ω − π + < 0.070 and $$ \mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{K}^{+}\right)/\mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{\pi}^{+}\right)<0.29. $$ B Ω c 0 → Ω − K + / B Ω c 0 → Ω − π + < 0.29 . 

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