Let G be a finite group. For k ∈ N, X ∈ Irr（G）, define c（k）x ：= 1/G ∑g∈G x（gk）. This is called the k-th Frobenius-Schur indicator of X. In this article we study the Probenius-Schur indicators for Frobenius ...Let G be a finite group. For k ∈ N, X ∈ Irr（G）, define c（k）x ：= 1/G ∑g∈G x（gk）. This is called the k-th Frobenius-Schur indicator of X. In this article we study the Probenius-Schur indicators for Frobenius groups, p-groups, semidihedral groups and mod- ular p-groups. Further, we use this to study the function ζ kG（g） which counts the number of roots of xk = g in a finite group G.展开更多
Let H be a semisimple Hopf algebra over an algebraically closed field Ik of characteristic p>dimk(H)^(1/2).We show that the antipode S of H satisfies the equality S^(2)(h)=uhu^(-1),where h e H,u=S(A_((2))A_((1))and...Let H be a semisimple Hopf algebra over an algebraically closed field Ik of characteristic p>dimk(H)^(1/2).We show that the antipode S of H satisfies the equality S^(2)(h)=uhu^(-1),where h e H,u=S(A_((2))A_((1))and A is a nonzero integral of H.The formula of s^(2) enables us to define higher Frobenius-Schur indicators for the Hopf algebra H.This generalizes the notion of higher Frobenius-Schur indicators from the case of characteristic O to the case of characteristic p>dimk(H)^(1/2).These indicators defined here share some properties with the ones defined over a field of characteristic 0.In particular,all these indicators are gauge invariants for the tensor category Rep(H)of finite-dimensional representations of H.展开更多
研究了单个线性关系是可闭线性关系的充分必要条件;并对L0=(A B C D),其中A,B,C,D是相应Hilbert空间上的线性关系,利用C相对A的有界性与B相对D的有界性及A,D的可闭性,推出了L_(0)也是可闭线性关系;同时,对于有界线性算子S=(S_(1) S_(2) ...研究了单个线性关系是可闭线性关系的充分必要条件;并对L0=(A B C D),其中A,B,C,D是相应Hilbert空间上的线性关系,利用C相对A的有界性与B相对D的有界性及A,D的可闭性,推出了L_(0)也是可闭线性关系;同时,对于有界线性算子S=(S_(1) S_(2) S_(3) S_(4)),得到当满足一定条件时L_(0)-μS的Frobenius-Schur分解公式,并得到了当L_(0)可闭时L_(0)的表达式,最后研究了L_(0)的S-本质谱。展开更多
首先研究了两个线性关系矩阵的乘积等于它们形式乘积的条件,在此基础上得到了线性关系矩阵L_(0)-μI=(A-μI B C D-μI)的Frobenius-Schur分解;其次利用Frobenius-Schur分解,讨论了L_(0)-μI和它的Schur补在单射情况、值域的稠密性以及...首先研究了两个线性关系矩阵的乘积等于它们形式乘积的条件,在此基础上得到了线性关系矩阵L_(0)-μI=(A-μI B C D-μI)的Frobenius-Schur分解;其次利用Frobenius-Schur分解,讨论了L_(0)-μI和它的Schur补在单射情况、值域的稠密性以及逆关系有界性之间的联系;最后刻画了L_(0)的点谱、剩余谱和连续谱。展开更多
In this paper,we propose a new approach towards the classification of spherical fusion categories by their Frobenius–Schur exponents.We classify spherical fusion categories of Frobenius–Schur exponent 2 up to monoid...In this paper,we propose a new approach towards the classification of spherical fusion categories by their Frobenius–Schur exponents.We classify spherical fusion categories of Frobenius–Schur exponent 2 up to monoidal equivalence.We also classify modular categories of Frobenius–Schur exponent 2 up to braided monoidal equivalence.It turns out that the Gauss sum is a complete invariant for modular categories of Frobenius–Schur exponent 2.This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.展开更多
In this paper, the adjoint of a densely defined block operator matrix L=[A B C D] in a Hilbert space X ×X is studied and the sufficient conditions under which the equality L＊=[A＊ B＊ C＊ D＊] holds are obtained...In this paper, the adjoint of a densely defined block operator matrix L=[A B C D] in a Hilbert space X ×X is studied and the sufficient conditions under which the equality L＊=[A＊ B＊ C＊ D＊] holds are obtained through applying Frobenius-Schur factorization.展开更多
文摘Let G be a finite group. For k ∈ N, X ∈ Irr（G）, define c（k）x ：= 1/G ∑g∈G x（gk）. This is called the k-th Frobenius-Schur indicator of X. In this article we study the Probenius-Schur indicators for Frobenius groups, p-groups, semidihedral groups and mod- ular p-groups. Further, we use this to study the function ζ kG（g） which counts the number of roots of xk = g in a finite group G.
基金National Natural Science Foundation of China(Grant No.12271243)National Natural Science Foundation of China(Grant No.12371041).
文摘Let H be a semisimple Hopf algebra over an algebraically closed field Ik of characteristic p>dimk(H)^(1/2).We show that the antipode S of H satisfies the equality S^(2)(h)=uhu^(-1),where h e H,u=S(A_((2))A_((1))and A is a nonzero integral of H.The formula of s^(2) enables us to define higher Frobenius-Schur indicators for the Hopf algebra H.This generalizes the notion of higher Frobenius-Schur indicators from the case of characteristic O to the case of characteristic p>dimk(H)^(1/2).These indicators defined here share some properties with the ones defined over a field of characteristic 0.In particular,all these indicators are gauge invariants for the tensor category Rep(H)of finite-dimensional representations of H.
文摘研究了单个线性关系是可闭线性关系的充分必要条件;并对L0=(A B C D),其中A,B,C,D是相应Hilbert空间上的线性关系,利用C相对A的有界性与B相对D的有界性及A,D的可闭性,推出了L_(0)也是可闭线性关系;同时,对于有界线性算子S=(S_(1) S_(2) S_(3) S_(4)),得到当满足一定条件时L_(0)-μS的Frobenius-Schur分解公式,并得到了当L_(0)可闭时L_(0)的表达式,最后研究了L_(0)的S-本质谱。
文摘首先研究了两个线性关系矩阵的乘积等于它们形式乘积的条件,在此基础上得到了线性关系矩阵L_(0)-μI=(A-μI B C D-μI)的Frobenius-Schur分解;其次利用Frobenius-Schur分解,讨论了L_(0)-μI和它的Schur补在单射情况、值域的稠密性以及逆关系有界性之间的联系;最后刻画了L_(0)的点谱、剩余谱和连续谱。
基金We thank Professor Siu-Hung Ng for helpful conversations.Z.Y.Wan is supported by the Shuimu Tsinghua Scholar Program.
文摘In this paper,we propose a new approach towards the classification of spherical fusion categories by their Frobenius–Schur exponents.We classify spherical fusion categories of Frobenius–Schur exponent 2 up to monoidal equivalence.We also classify modular categories of Frobenius–Schur exponent 2 up to braided monoidal equivalence.It turns out that the Gauss sum is a complete invariant for modular categories of Frobenius–Schur exponent 2.This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.
基金Supported by NSFC(Grant Nos.11101200,11371185,2013ZD01)
文摘In this paper, the adjoint of a densely defined block operator matrix L=[A B C D] in a Hilbert space X ×X is studied and the sufficient conditions under which the equality L＊=[A＊ B＊ C＊ D＊] holds are obtained through applying Frobenius-Schur factorization.
