antiautomorphism

antiautomorphism

[‚an·tē‚ȯd·ə′mȯr‚fiz·əm]
(mathematics)
An antiisomorphism of a ring, field, or integral domain with itself.
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be an involutive associative algebra antiautomorphism, and for x [member of] g define [(1 + x).sup.[dagger]] = 1 + [x.sup.[dagger]].
The supercharacter theory is also independent of the choice of subfield of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; that is, if F is any subfield of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [dagger] is an antiautomorphism of g when viewed as an F-algebra, we get the same supercharacter theory as by considering g as an [F.sub.q]-algebra.
Define an antiautomorphism [dagger] of [ut.sub.n]([F.sub.q]) by [x.sup.[dagger]] = J[x.sup.t]J.
The antiautomorphism [dagger] as defined above restricts to an antiautomorphism of [U.sub.P] for any symmetric poset.
Define an antiautomorphism [dagger] of [ut.sub.2n]([F.sub.q]) by [x.sup.[dagger]] = -[OMEGA][x.sup.[dagger]][OMEGA].
The map [x.sup.[dagger]] = J[[bar.x].sup.t]J defines an antiautomorphism of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if we consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as an [F.sub.q]-algebra.
Let D: R[([beta]).sub.x]-gmod [right arrow] [(R[([beta]).sub.x] -gmod).sup.opp] be the duality functor M [right arrow] [M.sup.*] induced by the antiautomorphism [psi] of R[([beta]).sub.x].
The ring [A.sup.(q).sub.n] is equipped with an involutive bar antiautomorphism defined by the Z-linear extension of [bar.[q.sup.1/2]] = [q.sup.-1/2] and [bar.[x.sub.ij]] = [x.sub.ij].
It is routine to check from the Relations 1-4 that [tau] extends to an involutive Z[[q.sup.[+ or -]1/2]]-algebra automorphism [tau] : [A.sup.(q).sub.n] [right arrow] [A.sup.(q).sub.n] and that [alpha] extends to an involutive Z[[q.sup.[+ or -]1/2]]-algebra antiautomorphism [alpha] : [A.sup.(q).sub.n] [right arrow] [A.sup.(q).sub.n].
Proof: Since the bar map is an antiautomorphism, we get that