اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe rank 8 case of a conjecture on square-zero upper triangular matrices
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تأليف
Berrin c{S}enturk
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Let $A$ be the polynomial algebra in $r$ variables with coefficients in an algebraically closed field $k$. When the characteristic of $k$ is $2$, Carlsson conjectured that for any $mathrm{dg}$-$A$-module $M$, which has dimension $N$ as a free $A$-module, if the homology of $M$ is nontrivial and finite dimensional as a $k$-vector space, then $Ngeq 2^r$. Here we examine a stronger conjecture concerning varieties of square-zero upper triangular $Ntimes N$ matrices with entries in $A$. Stratifying these varieties via Borel orbits, we show that the stronger conjecture holds when $N = 8$ without any restriction on the characteristic of $k$. This result also verifies that if $X$ is a product of $3$ spheres of any dimensions, then the elementary abelian $2$-group of order $4$ cannot act freely on $X$.
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In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian-Wachs conjecture, which links the Stanley-Stembridge conjecture in combinatorics to the geometry of Hessenberg varieties through Tymoczkos permutation group action on the
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A Banach space operator $Tin B({cal X})$ is polaroid if points $lambdainisosigmasigma(T)$ are poles of the resolvent of $T$. Let $sigma_a(T)$, $sigma_w(T)$, $sigma_{aw}(T)$, $sigma_{SF_+}(T)$ and $sigma_{SF_-}(T)$ denote, respectively, the approximat
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