Garrett 03-30-2012 1 • Interlude: Calculus on spheres: invariant integrals, invariant
... Theorem: (instance of Schur’s Lemma) For a finite-dimensional irreducible representation V of a group G, any G-intertwining ϕ : V → V of V to itself is scalar. Proof: First, claim that the collection HomG (V, V ) of all Gintertwinings of finite-dimensional V to itself is a division ring. Indeed, giv ...
... Theorem: (instance of Schur’s Lemma) For a finite-dimensional irreducible representation V of a group G, any G-intertwining ϕ : V → V of V to itself is scalar. Proof: First, claim that the collection HomG (V, V ) of all Gintertwinings of finite-dimensional V to itself is a division ring. Indeed, giv ...
Constructions of plane curves with many points
... The polynomial f (x) − f (y) now has the linear factor x − y. If we remove it, then the quotient is usually irreducible, has degree d = n−1, and has n2 −n = d2 + d roots (x, y) = (αi , αj ), αi 6= αj , which is slightly better in terms of the degree. If f is also assumed to be even, then f (x) − f ( ...
... The polynomial f (x) − f (y) now has the linear factor x − y. If we remove it, then the quotient is usually irreducible, has degree d = n−1, and has n2 −n = d2 + d roots (x, y) = (αi , αj ), αi 6= αj , which is slightly better in terms of the degree. If f is also assumed to be even, then f (x) − f ( ...
School of Mathematics and Statistics The University of Sydney
... 4. Z2 [y]y2 +1 - since y 2 +1 = (y+1)2 , this ring is the same as ring 3. There is an isomorphism from ring 3 to ring 4 defined by a + bx → a + b(y + 1). Check that. 5. Z2 [x]x2 +x = Z2 [x]x(x+1) ∼ = Z2 [x]x ⊕ Z2 [x]x+1 by the CRT - Since Z2 [x]x ∼ = Z2 and ...
... 4. Z2 [y]y2 +1 - since y 2 +1 = (y+1)2 , this ring is the same as ring 3. There is an isomorphism from ring 3 to ring 4 defined by a + bx → a + b(y + 1). Check that. 5. Z2 [x]x2 +x = Z2 [x]x(x+1) ∼ = Z2 [x]x ⊕ Z2 [x]x+1 by the CRT - Since Z2 [x]x ∼ = Z2 and ...
