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In [Indiana Univ. Math. J. 60 (2011), pp. 847-857] the first author introduced second order necessary conditions for a commuting square to admit sequential deformations in the moduli space of non-isomorphic commuting squares. In this paper we investigate these conditions for commuting squares CG constructed from finite groups G. We are especially interested in the case G = Zn, since deformations of CZn correspond to deformations of the Fourier matrix Fn in the moduli space of non-equivalent complex Hadamard matrices. We show that for G = Zn the second order conditions follow automatically from the first order conditions, but this is not necessarily true for other finite abelian groups G. Our result gives a complete description of the second order deformations of the Fourier matrix Fn in the moduli space of non-equivalent complex Hadamard matrices. [ABSTRACT FROM AUTHOR]

By a result of Schur [J. Reine Angew. Math. 140 (1911), pp. 1–28], the entrywise product M ∘ N of two positive semidefinite matrices M,N is again positive. Vybíral [Adv. Math. 368 (2020), p. 9] improved on this by showing the uniform lower bound M ∘ M ≥ En/n for all n × n real or complex correlation matrices M, where En is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 15 (1999), pp. 299–316] and to positive definite functions on groups. Vybíral (in his original preprint) asked if one can obtain similar uniform lower bounds for higher entrywise powers of M, or for M ∘ N when N ≠ M, M. A natural third question is to ask for a tighter lower bound that does not vanish as n → ∞, i.e., over infinite-dimensional Hilbert spaces. In this note, we affirmatively answer all three questions by extending and refining Vybíral's result to lower-bound M ∘ N, for arbitrary complex positive semidefinite matrices M,N. Specifically: we provide tight lower bounds, improving on Vybíral's bounds. Second, our proof is 'conceptual' (and self-contained), providing a natural interpretation of these improved bounds via tracial Cauchy–Schwarz inequalities. Third, we extend our tight lower bounds to Hilbert–Schmidt operators. As an application, we settle Open Problem 1 of Hinrichs–Krieg–Novak–Vybíral [J. Complexity 65 (2021), Paper No. 101544, 20 pp.], which yields improvements in the error bounds in certain tensor product (integration) problems. [ABSTRACT FROM AUTHOR]

We exhibit an operator norm bounded, infinite sequence \{A_n\} of 3n \times 3n complex matrices for which the commutator map X\mapsto XA_n - A_nX is uniformly bounded below as an operator over the space of trace-zero self-adjoint matrices equipped with Hilbert–Schmidt norm. The construction is based on families of quantum expanders. We give several potential applications of these matrices to the study of quantum expanders. We formulate several natural conjectures and provide numerical evidence. [ABSTRACT FROM AUTHOR]

The vector space of m x n complex matrices (m ⩾ n) admits a natural action of the group GL=GLm xGLn via row and column operations. For positive integers a, b, we consider the ideal Iaxb defined as the smallest GL-equivariant ideal containing the b-th powers of the a x a minors of the generic m x n matrix. We compute the syzygies of the ideals Iaxb for all a, b, together with their GL-equivariant structure, generalizing earlier results of Lascoux for the ideals of minors (b=1) and of Akin-Buchsbaum-Weyman for the powers of the ideals of maximal minors (a=n). Our methods rely on a nice connection between commutative algebra and the representation theory of the superalgebra gl(mǀn), as well as on our previous calculation of Ext modules done in the context of describing local cohomology with determinantal support. Our results constitute an important ingredient in the proof by Nagpal-Sam-Snowden of the first non-trivial Noetherianity results for twisted commutative algebras which are not generated in degree one. [ABSTRACT FROM AUTHOR]