Word Measures on Unitary Groups
نویسندگان
چکیده
We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups U (n). Every word w in the free group Fr on r generators determines a word map from U (n) to U (n), defined by substitutions. The w-measure on U (n) is defined as the pushforward via this word map of the Haar measure on U (n). Let T rw (n) denote the expected trace of a random unitary matrix sampled from U (n) according to the w-measure. It was shown by Voiculescu [Voi91] that for w 6= 1 this expected trace is o (n) asymptotically in n. We relate the numbers T rw (n) to the theory of commutator length of words and obtain a much stronger statement: T rw (n) = O ( n1−2g ) , where g is the commutator length of w. Moreover, we analyze the number limn→∞ n2g−1 · T rw (n) and show it is an integer which, roughly, counts the number of (equivalence classes of) solutions to the equation [u1, v1] . . . [ug, vg] = w with ui, vi ∈ Fr. Similar results are obtained for finite sets of words and their commutator length, and we deduce that one can “hear” the stable commutator length of a word by “listening” to its unitary measures.
منابع مشابه
On local gamma factors for orthogonal groups and unitary groups
In this paper, we find a relation between the proportionality factors which arise from the functional equations of two families of local Rankin-Selberg convolutions for irreducible admissible representations of orthogonal groups, or unitary groups. One family is that of local integrals of the doubling method, and the other family is that of local integrals expressed in terms of sph...
متن کاملSome bounds on unitary duals of classical groups - non-archimeden case
We first give bounds for domains where the unitarizabile subquotients can show up in the parabolically induced representations of classical $p$-adic groups. Roughly, they can show up only if the central character of the inducing irreducible cuspidal representation is dominated by the square root of the modular character of the minimal parabolic subgroup. For unitarizable subquotients...
متن کاملBehavior of $R$-groups for $p$-adic inner forms of quasi-split special unitary groups
We study $R$-groups for $p$-adic inner forms of quasi-split special unitary groups. We prove Arthur's conjecture, the isomorphism between the Knapp-Stein $R$-group and the Langlands-Arthur $R$-group, for quasi-split special unitary groups and their inner forms. Furthermore, we investigate the invariance of the Knapp-Stein $R$-group within $L$-packets and between inner forms. This w...
متن کاملThe Uniform Word Problem for Groups and Finite Rees Quotients of E-unitary Inverse Semigroups
متن کامل
Gaussian elimination in split unitary groups with an application to public - key cryptography ∗
Gaussian elimination is used in special linear groups to solve the word problem. In this paper, we extend Gaussian elimination to split unitary groups. These algorithms have an application in building a public-key cryptosystem, we demonstrate that. 2010 MSC: 20H30, 94A60
متن کامل