On the realization of the Gelfand character of a finite group as a twisted trace

We show that the Gelfand character χ G \chi_{G} of a finite group 𝐺 (i.e. the sum of all irreducible complex characters of 𝐺) may be realized as a “twisted trace” g ↦ Tr ⁡ ( ρ g ∘ T ) g\mapsto\operatorname{Tr}(\rho_{g}\circ T) for a suitable involutive linear automorphism 𝑇 of L 2 ⁢ ( G ) L^{2}(G) , where ( L 2 ⁢ ( G ) , ρ ) (L^{2}(G),\rho) is the right regular representation of 𝐺. Moreover, we prove that, under certain hypotheses, we have T ⁢ ( f ) = f ∘ L T(f)=f\circ L ( f ∈ L 2 ⁢ ( G ) f\in L^{2}(G) ), where 𝐿 is an involutive anti-automorphism of 𝐺. The natural representation 𝜏 of 𝐺 associated to the natural 𝐿-conjugacy action of 𝐺 in the fixed point set Fix G ⁡ ( L ) \operatorname{Fix}_{G}(L) of 𝐿 turns out to be a Gelfand model for 𝐺 in some cases. We show that ( L 2 ⁢ ( Fix G ⁡ ( L ) ) , τ ) (L^{2}(\operatorname{Fix}_{G}(L)),\tau) fails to be a Gelfand model if 𝐺 admits non-trivial central involutions.

Linear sets and MRD-codes arising from a class of scattered linearized polynomials

A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the q-polynomial over $${{\mathbb {F}}}_{q^6}$$ F q 6 , $$q \equiv 1\pmod 4$$ q ≡ 1 ( mod 4 ) described in Bartoli et al. (ARS Math Contemp 19:125–145, 2020) and Zanella and Zullo (Discrete Math 343:111800, 2020) is generalized for any even $$n\ge 6$$ n ≥ 6 to an $${{{\mathbb {F}}}_q}$$ F q -linear automorphism $$\psi (x)$$ ψ ( x ) of $${{\mathbb {F}}}_{q^n}$$ F q n of order n. Such $$\psi (x)$$ ψ ( x ) and some functional powers of it are proved to be scattered. In particular, this provides new maximum scattered linear sets of the projective line $${{\,\mathrm{{PG}}\,}}(1,q^n)$$ PG ( 1 , q n ) for $$n=8,10$$ n = 8 , 10 . The polynomials described in this paper lead to a new infinite family of MRD-codes in $${{\mathbb {F}}}_q^{n\times n}$$ F q n × n with minimum distance $$n-1$$ n - 1 for any odd q if $$n\equiv 0\pmod 4$$ n ≡ 0 ( mod 4 ) and any $$q\equiv 1\pmod 4$$ q ≡ 1 ( mod 4 ) if $$n\equiv 2\pmod 4$$ n ≡ 2 ( mod 4 ) .

Galois Involutions

This chapter focuses on the fixed points of a strictly semi-linear automorphism of order 2 of a spherical building which satisfies the conditions laid out in Hypothesis 30.1. It begins with the fhe definition of a spherical building satisfying the Moufang condition and a Galois involution of Δ‎, described as an automorphism of Δ‎ of order 2 that is strictly semi-linear. It can be recalled that Δ‎ can have a non-type-preserving semi-linear automorphism only if its Coxeter diagram is simply laced. The chapter assumes that the building Δ‎ being discussed is as in 30.1 and that τ‎ is a Galois involution of Δ‎. It also considers the notation stating that the polar region of a root α‎ of Δ‎ is the unique residue of Δ‎ containing the arctic region of α‎.

Strictly Semi-linear Automorphisms

This chapter considers the action of a strictly semi-linear automorphism fixing a root on the corresponding root group. It begins with the hypothesis whereby Δ‎ is a Moufang spherical building and Π‎ is the Coxeter diagram of Δ‎; here the chapter fixes an apartment Σ‎ of Δ‎ and a root α‎ of Σ‎. The discussion then turns to a number of assumptions about an isomorphism of Moufang sets, anisotropic quadratic space, and root group sequence, followed by a lemma where E is an octonion division algebra with center F and norm N and D is a quaternion subalgebra of E. The chapter concludes with three versions of what is really one result about fixed points of non-linear automorphisms of the Moufang sets.

Linear Automorphisms

This chapter considers the notion of a linear automorphism of an arbitrary spherical building satisfying the Moufang property. It begins with the notation whereby Ω‎ = (U₊, U₁, ..., Uₙ) is the root group sequence and x₁, ... , xₙ the isomorphisms obtained by applying the recipe in [60, 16.x] for x = 1, 2, 3, ... or 9 to a parameter system Λ‎ of the suitable type (and for suitable n) and Δ‎ is the corresponding Moufang n-gon. The chapter proceeds by looking at cases where Λ‎ is a proper anisotropic pseudo-quadratic space defined over an involutory set or a quadratic space of type E⁶, E₇ or E₈. It also describes a notation dealing with the Moufang spherical building with Coxeter diagram Λ‎, an apartment of Δ‎, and a chamber of Σ‎.

On the extension of unitary group isomorphisms of unital UHF-algebras

H. Dye showed that an isomorphism between the (discrete) unitary groups in two factors not of type In is implemented by a linear (or a conjugate linear) *-isomorphism of the factors. If φ is an isomorphism between the unitary groups of two unital C*-algebras, it induces a bijective map θφ between the sets of projections. For certain UHF-algebras, we construct an automorphism φ of their unitary group, such that θφ does not preserve the orthogonality of projections. For a large class of unital finite C*-algebras, we show that θφ is always an orthoisomorphism. If φ is a continuous automorphism of the unitary group of a UHF-algebra A, we show that φ is implemented by a linear or a conjugate linear *-automorphism of A.

Zariski Hyperplane Section Theorem for Grassmannian Varieties

Let ϕ: X → M be a morphism from a smooth irreducible complex quasi-projective variety X to a Grassmannian variety M such that the image is of dimension ≥ 2. Let D be a reduced hypersurface in M, and γ a general linear automorphism of M. We show that, under a certain differentialgeometric condition on ϕ(X) and D, the fundamental group π1((γ ○ ϕ)−1 (M \ D)) is isomorphic to a central extension of π1(M \ D) × π1(X) by the cokernel of π2(ϕ) : π2(X) → π2(M).

Automorphisms of G-Azumaya Algebras

Let R be a commutative ring, G a finite abelian group of order n and exponent m, and assume n is a unit in R. In [10], F. W. Long defined a generalized Brauer group, BD(R, G), of algebras with a G-action and G-grading, whose elements are equivalence classes of G-Azumaya algebras. In this paper we investigate the automorphisms of a G-Azumaya algebra A and prove that if Picm(R) is trivial, then these automorphisms are all, in some sense, inner.In fact, each of these “inner” automorphisms can be written as the composition of an inner automorphism in the usual sense and a “linear“ automorphism, i.e., an automorphism of the typewith r(σ) a unit in R. We then use these results to show that the group of gradings of the centre of a G-Azumaya algebra A is a direct summand of G, and thus if G is cyclic of order pr, A is the (smash) product of a commutative and a central G-Azumaya algebra.

The orbit of a Hölder continuous path under a hyperbolic toral automorphism

Let f:T3→T3 be a hyperbolic toral automorphism lifting to a linear automorphism with real eigenvalues. We prove that there is a Hölder continuous path in T3 whose orbit-closure is 1-dimensional. This strengthens results of Hancock and Przytycki concerning continuous paths, and contrasts with results of Franks and Mañé concerning rectifiable paths.