# Blow-up and nonexistence of sign changing solutions to the by Ben Ayed M., El Mehdi K., Pacella F. PDF

By Ben Ayed M., El Mehdi K., Pacella F.

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2) hold for a complex conjugate K. 1) and is anti-unitary. 1) holds for J anti-unitary if and only if U is self-conjugate. 1) holds and let J2 = L. Then L is unitary and LU(x) = J 2 U(x) = JU(x)J = U(x)J 2 = U(x)L, so by Schur's lemma, L =ell. Since Lis unitary, c = ei 0 . But L commutes with J so J(cll) = cliJ; that is, c = c so c = ±1. 1), J 1 J 2 =ell or J 1 = ±cJ2 , so J is unique up to phase. 3 below, there is an orthonormal basis where J 'Pi = 'Pi and so (cpi, U(x)cpi) = (JUcpj, Jcpi) = (U Jcpi, Jcpi) = (Ucpi, 'Pi)= (cpi, U(x)cpi), so the matrix elements of U are real.

3. Any representation can be written as a direct sum of irreps. Proof. We'll prove this by induction on deg(U), the degree of U (= dim(X)). If deg(U) = 1, U is irreducible since any one-dimensional representation is irreducible. Suppose we have the result for all representations of degree< d. Let deg(U) =d. If U is irreducible, we can stop. If not, U = U1 EBU2 with deg(U1) < d, deg(U2 ) < d; so by induction, each Ui is a direct sum of irreducibles and so U is. D This says any U can be written U1 EB · · · EB Uk with each Uk an irrep.

This is a normal subgroup since Z( G) is the center of G. Let UAa) be the representation of an given by on 'Ha 0 ... 0 'Ha. We claim UAa) is irreducible; for example, its character obeys Note that UAa) is distinct from the natural tensor product representation of G, 0nU(a); it is related by while u\$,a) is irreducible, 0nu(a) may not be. By Schur's lemma, if z E Z(G), then U(a)(z) = q(z)ll, where q is a onedimensional representation of Z. Thus if (g 1 , ... , 9n) E Tn, we have uj,al(g1, ... ,gn) = q(gi)ll0 ...