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In this paper, we perform a systematic study of permutation statistics and bijective maps on permutations in which we identify and prove 122 instances of the homomesy phenomenon. Homomesy occurs when the average value of a statistic is the same on each orbit of a given map. The maps we investigate include the Lehmer code rotation, the reverse, the complement, the Foata bijection, and the Kreweras complement. The statistics studied relate to familiar notions such as inversions, descents, and permutation patterns, and also more obscure constructs. Besides the many new homomesy results, we discuss our research method, in which we used SageMath to search the FindStat combinatorial statistics database to identify potential homomesies. [ABSTRACT FROM AUTHOR]

Let \mathcal M be a semifinite factor with a faithful normal semifinite tracial weight \tau, and \mathscr P the set of all projections in \mathcal M. Denote by \mathscr P_{c} the Grassmann space of all projections in \mathscr P with trace c, where c is a positive real number. A map \psi : \mathscr P_c\rightarrow \mathscr P_c is called an ortho-isomorphism if \psi is a bijection of \mathscr P_c onto \mathscr P_c satisfying, for all P,Q\in \mathscr P_c, P\perp Q if and only if \psi (P)\perp \psi (Q). The aim of this paper is to establish a version of Uhlhorn's theorem in the setting of semifinite factors. We give a complete characterization of ortho-isomorphisms on Grassmann space \mathscr P_c in a semifinite factor. And we show that an ortho-isomorphism \psi : \mathscr P_c\rightarrow \mathscr P_c can be extended to a Jordan *-isomorphism \rho of \mathcal M onto \mathcal M. As an application, we obtain the structure of surjective isometries on \mathscr P_c with respect to strictly increasing unitarily invariant norms. [ABSTRACT FROM AUTHOR]

In [Math. Ann. 367 (2017), pp. 1517–1586] Bar-Natan and the first author show that solutions to the Kashiwara–Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in \mathbb {R}^4, which can be finitely presented algebraically as a circuit algebra , or equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups \mathsf {KV} and \mathsf {KRV}—the symmetry groups of Kashiwara-Vergne solutions—as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebras of arrow diagrams, respectively. Finally, we provide a description of the graded Grothendieck-Teichmüller group \mathsf {GRT}_1 as automorphisms of arrow diagrams. [ABSTRACT FROM AUTHOR]

Webs are certain planar diagrams embedded in disks. They index and describe bases of tensor products of representations of \mathfrak {sl}_2 and \mathfrak {sl}_3. There are explicit bijections between webs and certain rectangular tableaux. Work of Petersen–Pylyavskyy–Rhoades (2009) and Russell (2013) shows that these bijections relate web rotation to tableau promotion. We describe the analogous relation between web reflection and tableau evacuation. [ABSTRACT FROM AUTHOR]

An extended derivation (endomorphism) of a (restricted) Lie algebra L is an assignment of a derivation (respectively) of L' for any (restricted) Lie morphism f:L\to L', functorial in f in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of L' to every f; and (b) if L is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then L is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms. [ABSTRACT FROM AUTHOR]

Given a conic \mathcal {C} over \mathbb {Q}, it is natural to ask what real points on \mathcal {C} are most difficult to approximate by rational points of low height. For the analogous problem on the real line (for which the least approximable number is the golden ratio, by Hurwitz's theorem), the approximabilities comprise the classically studied Lagrange and Markoff spectra, but work by Cha–Kim [ Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum , arXiv: 1903.02882, 2021] and Cha–Chapman–Gelb–Weiss [Monatsh. Math. 197 (2022), pp. 1–55] shows that the spectra of conics can vary. We provide notions of approximability, Lagrange spectrum, and Markoff spectrum valid for a general \mathcal {C} and prove that their behavior is exhausted by the special family of conics \mathcal {C}_n : XZ = nY^2, which has symmetry by the modular group \Gamma _0(n) and whose Markoff spectrum was studied in a different guise by A. Schmidt and Vulakh [Trans. Amer. Math. Soc. 352 (2000), pp. 4067–4094]. The proof proceeds by using the Gross-Lucianovic bijection to relate a conic to a quaternionic subring of \operatorname {Mat}^{2\times 2}(\mathbb {Z}) and classifying invariant lattices in its 2-dimensional representation. [ABSTRACT FROM AUTHOR]

Starting from an integrable rank-n vertex model, we construct an explicit family of partition functions indexed by compositions \mu = (\mu _1,\dots,\mu _n). Using the Yang–Baxter algebra of the model and a certain rotation operation that acts on our partition functions, we show that they are eigenfunctions of the Cherednik–Dunkl operators Y_i for all 1 \leqslant i \leqslant n, and are thus equal to nonsymmetric Macdonald polynomials E_{\mu }. Our partition functions have the combinatorial interpretation of ensembles of coloured lattice paths which traverse a cylinder. Applying a simple bijection to such path ensembles, we show how to recover the well-known combinatorial formula for E_{\mu } due to Haglund–Haiman–Loehr. [ABSTRACT FROM AUTHOR]

We study the effect of the Howe correspondence on Harish-Chandra series for type I dual pairs over finite fields with odd characteristic. We define a bijection obtained from this correspondence, and enjoying the property of "having minimal unipotent support". Finally, we examine the interaction between the Howe correspondence and weak cuspidality. [ABSTRACT FROM AUTHOR]

We provide a short and elementary proof of the main technical result of the recent article "Uniqueness of Clifford torus with prescribed isoperimetric ratio" by Thomas Yu and Jingmin Chen [Proc. Amer. Math. Soc. 150 (2022), pp. 1749–1765]. The key of the new proof is an explicit expression of the central function (\mathsf {Iso}, to be proved bijective) as a quotient of Gaussian hypergeometric functions. [ABSTRACT FROM AUTHOR]

For a prime ℓ, the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with ℓ'-degree and the corresponding set for the normalizer of a Sylow ℓ-subgroup. Navarro's refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro–Späth–Vallejo have reduced the McKay–Navarro conjecture to certain "inductive" conditions on finite simple groups. We prove that these inductive McKay–Navarro (also called the inductive Galois–McKay) conditions hold for the prime ℓ =2 for several groups of Lie type, namely the untwisted groups without non-trivial graph automorphisms. [ABSTRACT FROM AUTHOR]

We investigate an invariant for continuous fields of the Cuntz algebra \mathcal {O}_{n+1} introduced by Taro Sogabe [Math. Ann. 380 (2021), pp. 91–117], and find a way to obtain a continuous field of \mathbb {M}_n(\mathcal {O}_\infty) from that of \mathcal {O}_{n+1} using the construction of the invariant. By Brown's representability theorem, this gives a bijection from the set of the isomorphism classes of continuous fields of \mathcal {O}_{n+1} to those of \mathbb {M}_n(\mathcal {O}_\infty). As a consequence, we obtain a new proof for M. Dadarlat's classification result of continuous fields of \mathcal {O}_{n+1} arising from vector bundles, which corresponds to those of \mathbb {M}_n(\mathcal {O}_\infty) stably isomorphic to the trivial field. [ABSTRACT FROM AUTHOR]