Symmetric functions of two noncommuting variables Nicholas Young Leeds and Newcastle Universities Joint work with Jim Agler, UCSD Newcastle, November 013
Abstract We prove a noncommutative analogue of the fact that every symmetric analytic function of (z, w) in the bidisc D can be expressed as an analytic function of the variables z + w and zw. We construct an analytic nc-map π from the biball to an infinite-dimensional nc-domain Ω with the property that, for every bounded symmetric analytic function ϕ of two noncommuting variables on the biball, there exists a bounded analytic nc-function Φ on Ω such that ϕ = Φ π. We also establish a realization formula for Φ, and hence for ϕ, in terms of operators on Hilbert space.
Symmetric polynomials Every symmetric polynomial in two commuting variables z and w can be written as a polynomial in the variables z + w and zw. What if z and w do not commute? The polynomial zwz + wzw in non-commuting variables z, w cannot be written as p(z + w, zw + wz) for any nc-polynomial p.
Proof Suppose that zwz + wzw = f(z + w, zw + wz) for some polynomial f and that f(x, 0) = j f j x j. Choose [ ] [ ] 1 0 0 0 z =, w = ; then 0 1 1 0 zw + wz = 0, zwz + wzw = [ ] 0 0, z + w = 1 0 [ ] 1 0. 1 1 Note that (z + w) = 1. Hence [ ] [ ] j 0 0 1 0 f 1 0 j = A1 1 1 + B = j [ ] 1 0 1 1 for some A, B C, which is impossible.
Theorem (Margarete Wolf, 1936) There is no finite basis for the ring of symmetric polynomials in d nc indeterminates over C when d > 1. Here nc means noncommuting or noncommutative. Polynomial nc-functions Let M n denote the space of complex n n complex matrices. If f is a polynomial in nc indeterminates x 1,..., x d then f determines a map f : n=1 (M n ) d M n n=1 by substitution, as in the previous example. Poincaré: f = g if and only if f = g.
Properties of polynomial nc functions Regard the space M d def = n=1 (M n ) d as the nc analogue of C d. Endow M d with the discrete union topology. If f is a polynomial in d nc indeterminates then f satisfies 1) f maps (M n ) d to M n for each n 1; ) f (x y) = f (x) f (y) for all x, y M d ; 3) f (s 1 xs) = s 1 f (x)s for all n 1 and all s M n,, x (M n ) d with s invertible. Here s 1 (x 1,..., x d )s means (s 1 x 1 s,..., s 1 x d s).
J. L. Taylor, 1974. Analytic nc functions Let Ω M d be an nc domain. An analytic nc function on Ω is a map f : Ω M 1 such that 1) f maps Ω (M n ) d analytically to M n ( f is graded ); ) f(x y) = f(x) f(y) for all x, y Ω ( f respects direct sums ); 3) f(s 1 xs) = s 1 f(x)s for all x Ω and all invertible s such that s 1 xs Ω ( f respects similarities ). An nc domain is an open subset of M d that is closed under direct sums and unitary similarity.
Symmetric analytic functions on the bidisc Let π : C C be given by π(z, w) = (z + w, zw). If ϕ : D C is analytic and symmetric in z and w then there exists an analytic function Φ : π(d ) C such that ϕ = Φ π: D ϕ π π(d ) C Φ Are there analogous statements for symmetric functions of non-commuting variables? What is the analogue of π(d )?
The biball B is the non-commutative analogue of the bidisc: B def = n=1 B n B n M where B n denotes the open unit ball of the space M n of n n complex matrices. It is an nc-domain: 1) it is open in n=1 M n; ) if x = (x 1, x ) B and y = (y 1, y ) B then x y def = (x 1 y 1, x y ) B ; 3) if x B M n and u is an n n unitary then (u x 1 u, u x u) B.
Symmetric analytic nc-functions on B There is an nc-domain Ω in the space M def = M n n=1 and an analytic nc-map π : B Ω (given by a simple rational expression) such that every bounded symmetric analytic nc-function ϕ on B factors through π, and conversely. B ϕ π n M n Φ Φ can be expressed by means of a non-commutative version of the familiar linear fractional realization formula for functions in the Schur class. Ω
Theorem There exists an nc-domain Ω in M such that where π : B M which maps x (u, v, vuv, vu v,... ), u = x1 + x, v = x1 x, has the following three properties. 1) π is an analytic nc-map from B to Ω; ) for every bounded symmetric analytic nc-function ϕ on the biball there exists an analytic nc-function Φ on Ω such that ϕ = Φ π; 3) for every g Ω and every contraction T the operator g(t ) exists.
Ω and π Ω M is the open unit ball of the nc disc algebra. A sequence (g 0, g 1, g,... ) M n belongs to Ω if and only if g(z) = g 0 + g 1 z + g z +... is the Taylor series of an n n-matrix function g that is analytic in the open unit disc D and continous on the closure of D, and moreover satisfies sup z D g(z) < 1. Since π(x) = (u, v, vuv, vu v,... ), π(x)(z) = u + v z + vuvz + vu vz 3 +... = u + vz(1 + uz + u z +... )v = u + vz(1 uz) 1 v = x1 + x + x1 x z ( 1 x1 + x z ) 1 x 1 x.
The functional calculus Let g = (g 0, g 1, g,... ) M n. power series j=0 g j z j. Identify g with the formal Then, for any matrix or bounded linear operator T we may write down the series j=0 g j T j. If T acts on a linear space H then the sum g(t ) of the series, if the series converges in an appropriate topology, is an operator on C n H. The map g g(t ) is called the functional calculus; it is defined (for a given operator T ) on a subset of M. 3) says: π(x)(t ) is defined [converges] for any x B and any operator T such that T 1.
A realization formula For any symmetric analytic nc-function ϕ on the biball there exist a unitary operator U on l and a contractive operator [ ] a B : C l C l C D such that the function Φ on Ω defined, for n 1 and g Ω M n, by Φ(g) = a1 n + (1 n B)g(U) (1 (1 n D)g(U)) 1 (1 n C) is an nc-function satisfying ϕ = Φ π.
Another formulation Identify a sequence (g j ) M with the formal power series j 0 g j z j. Then π(x)(z) = x1 + x + x1 x ( z 1 x1 + x ) 1 x 1 x z. For x B M n and a unitary operator U on l, π(x)(u) = u 1 + v U + vuv U +... = x1 + x ( x 1 x ) 1 l + U ( 1 C n l x1 + x ) 1 ( x 1 x U an operator on C n l. 1 l ),
Realization again Let ϕ be a symmetric nc-function on B bounded by 1 in norm. There exist a unitary U on l and a contraction [ ] a B : C l C l C D such that, for x B M n, where ϕ(x) = a1 n + (1 n B)π(x)(U) (1 (1 n D)π(x)(U)) 1 (1 n C) π(x)(u) = x1 + x ( 1 l + 1 C n l x1 + x ( x 1 x U U ) ) 1 ( x 1 x 1 l ).
A final question about symmetric polynomials The polynomials u, v, vuv, vu v,... constitute a basis for the algebra of symmetric polynomials in x 1, x, where Observe that u = 1 (x1 + x ), v = 1 (x1 x ). vu v = vuv(v ) 1 vuv, vu 3 v = vuv(v ) 1 vuv(v ) 1 vuv,... Hence the algebra of rational symmetric functions in two variables has a basis consisting of the three functions u, v, vuv. Is there an analogue for symmetric rational functions in three variables?
Reference Symmetric functions of two noncommuting variables, by Jim Agler and N. J. Young, arxiv:1307.1588