Hecke algebras
... Knowing the structure of the Hecke algebra H(G//B) is only a first basic step. Understanding the decomposition of C[B\G] as a representation of G requires much more, eventually the theory of [Kazhdan-Lusztig:1979]. Similar algebras, called Iwahori-Hecke algebras, arise in the theory of unramified re ...
... Knowing the structure of the Hecke algebra H(G//B) is only a first basic step. Understanding the decomposition of C[B\G] as a representation of G requires much more, eventually the theory of [Kazhdan-Lusztig:1979]. Similar algebras, called Iwahori-Hecke algebras, arise in the theory of unramified re ...
Efficient Diffie-Hellman Two Party Key Agreement
... G together with an operation * defined on pairs of elements of G; The order of the group is the number of elements in G. The operation must have certain properties, similar to those with which we are familiar from ordinary integer arithmetic. For example, the integers modulo n, namely =n = { 0, 1, 2 ...
... G together with an operation * defined on pairs of elements of G; The order of the group is the number of elements in G. The operation must have certain properties, similar to those with which we are familiar from ordinary integer arithmetic. For example, the integers modulo n, namely =n = { 0, 1, 2 ...
arXiv:math.OA/9901094 v1 22 Jan 1999
... cocycle functor ZΓ : Ab(Γ) → Ab, where Ab(Γ) is the category of Γ-sheaves and Ab is the category of abelian groups, is defined as follows: Given a Γ-sheaf A, the abelian group ZΓ (A) consists of all continuous functions f : Γ → A such that f (γ) ∈ Ar(γ) (i.e. f is a continuous section of r∗ (A)) and ...
... cocycle functor ZΓ : Ab(Γ) → Ab, where Ab(Γ) is the category of Γ-sheaves and Ab is the category of abelian groups, is defined as follows: Given a Γ-sheaf A, the abelian group ZΓ (A) consists of all continuous functions f : Γ → A such that f (γ) ∈ Ar(γ) (i.e. f is a continuous section of r∗ (A)) and ...
pdf
... matrices of determinant one having entries from the nite eld Fq of q elements. The projective special linear group PSL2 (Z=qZ) is obtained by dividing SL2 (Z=qZ) by its center, fI g where I is the 2 2 identity matrix, and is a simple nite group of Lie type (for q 5). The group PSL2 (Z=qZ) ha ...
... matrices of determinant one having entries from the nite eld Fq of q elements. The projective special linear group PSL2 (Z=qZ) is obtained by dividing SL2 (Z=qZ) by its center, fI g where I is the 2 2 identity matrix, and is a simple nite group of Lie type (for q 5). The group PSL2 (Z=qZ) ha ...
pdf file on-line
... C : Fin → Algf from the category of finite spaces to the category of finite-dimensional ∗-algebras. Indeed, a map φ : X1 → X2 of finite spaces induces a map φ∗ : C(X2 ) → C(X1 ) by pullback: φ∗ f = φ ◦ f ∈ C(X1 ) when f ∈ C(X2 ). We arrive at the following Question: Can we ‘invert’ the functor C? In ...
... C : Fin → Algf from the category of finite spaces to the category of finite-dimensional ∗-algebras. Indeed, a map φ : X1 → X2 of finite spaces induces a map φ∗ : C(X2 ) → C(X1 ) by pullback: φ∗ f = φ ◦ f ∈ C(X1 ) when f ∈ C(X2 ). We arrive at the following Question: Can we ‘invert’ the functor C? In ...
Representations with Iwahori-fixed vectors
... • Irreducibility criteria Using the ideas of [Casselman 1980] descended from the Borel-Matsumoto theorem on admissible representations of p-adic reductive groups containing Iwahori-fixed vectors, it is possible to give an easily verifiable sufficient criterion for irreducibility of degenerate princi ...
... • Irreducibility criteria Using the ideas of [Casselman 1980] descended from the Borel-Matsumoto theorem on admissible representations of p-adic reductive groups containing Iwahori-fixed vectors, it is possible to give an easily verifiable sufficient criterion for irreducibility of degenerate princi ...
Computing Galois groups by specialisation
... where each generator fixes those of {t1 , t2 , t3 , 2, i} not mentioned. The multiplication in G is as follows: all the σi commute with each other, and τ σi = σi−1 τ . Let a = (a1 , a2 , a3 ) be a Q-valued point of U . Then π −1 a consists of finitely many points of V , which are permuted by the act ...
... where each generator fixes those of {t1 , t2 , t3 , 2, i} not mentioned. The multiplication in G is as follows: all the σi commute with each other, and τ σi = σi−1 τ . Let a = (a1 , a2 , a3 ) be a Q-valued point of U . Then π −1 a consists of finitely many points of V , which are permuted by the act ...
Existence of almost Cohen-Macaulay algebras implies the existence
... result of this paper. Theorem 1 (Theorem 3.2). For a complete Noetherian local domain, if it is contained in an almost Cohen-Macaulay domain, then there exists a balanced big Cohen-Macaulay algebra over it. In [8], Hochster proves the existence of weakly functorial big Cohen-Macaulay algebras from t ...
... result of this paper. Theorem 1 (Theorem 3.2). For a complete Noetherian local domain, if it is contained in an almost Cohen-Macaulay domain, then there exists a balanced big Cohen-Macaulay algebra over it. In [8], Hochster proves the existence of weakly functorial big Cohen-Macaulay algebras from t ...
Non-standard number representation: computer arithmetic, beta
... Non-standard number representation is emerging as a new research field, with many difficult open questions, and several important applications. The notions presented in this contribution are strongly related to the chapters of this volume written by Akiyama, Pelantová and Masáková, and Sakarovitch. ...
... Non-standard number representation is emerging as a new research field, with many difficult open questions, and several important applications. The notions presented in this contribution are strongly related to the chapters of this volume written by Akiyama, Pelantová and Masáková, and Sakarovitch. ...
Algebra for Digital Communication
... (3) Let’s give explicit descriptions of these two homomorphisms, constructed, as usual, by sending [1] (in Z/4Z or Z/12Z) on 1R = [9]12 . Then using additivity, the only possibility is: f ([x]4 ) = f (x · [1]4 ) = x · f ([1]4 ) = x · [9]12 = [9x]12 , and g([x]12 ) = [9x]12 . We can then verify that ...
... (3) Let’s give explicit descriptions of these two homomorphisms, constructed, as usual, by sending [1] (in Z/4Z or Z/12Z) on 1R = [9]12 . Then using additivity, the only possibility is: f ([x]4 ) = f (x · [1]4 ) = x · f ([1]4 ) = x · [9]12 = [9x]12 , and g([x]12 ) = [9x]12 . We can then verify that ...
s13 - Math-UMN
... [13.9] Find the irreducible factors of x5 − 4 in Q[x]. In Q(ζ)[x] with a primitive fifth root of unity ζ. First, by Eisenstein’s criterion, x5 −2 is irreducible over Q, so the fifth root of 2 generates a quintic extension of Q. Certainly a fifth root of 4 lies in such an extension, so must be either ...
... [13.9] Find the irreducible factors of x5 − 4 in Q[x]. In Q(ζ)[x] with a primitive fifth root of unity ζ. First, by Eisenstein’s criterion, x5 −2 is irreducible over Q, so the fifth root of 2 generates a quintic extension of Q. Certainly a fifth root of 4 lies in such an extension, so must be either ...
20. Cyclotomic III - Math-UMN
... unity are expressible as ζ a with a in ( /n)× . More precisely, we saw earlier that for any other root β of f (x) = 0 in (α) with f the minimal polynomial of α over , there is an automorphism of (α) sending α to β. Thus, for any a relatively prime to n there is an automorphism which sends ζ −→ ζ a . ...
... unity are expressible as ζ a with a in ( /n)× . More precisely, we saw earlier that for any other root β of f (x) = 0 in (α) with f the minimal polynomial of α over , there is an automorphism of (α) sending α to β. Thus, for any a relatively prime to n there is an automorphism which sends ζ −→ ζ a . ...
Posets 1 What is a poset?
... An element x is an upper bound for a subset Y of X if y ≤R x for all y ∈ Y . Lower bounds are defined similarly. We say that x is a least upper bound or l.u.b. of Y if it is an upper bound and satisfies x ≤R x0 for any upper bound x0 . The concept of a greatest lower bound or g.l.b. is defined simil ...
... An element x is an upper bound for a subset Y of X if y ≤R x for all y ∈ Y . Lower bounds are defined similarly. We say that x is a least upper bound or l.u.b. of Y if it is an upper bound and satisfies x ≤R x0 for any upper bound x0 . The concept of a greatest lower bound or g.l.b. is defined simil ...
1. Affinoid algebras and Tate`s p-adic analytic spaces : a brief survey
... Definition 1.15. A rigid analytic space is a locally ringed G-space (X, T , O) admitting a covering {Ui } ∈ Cov(X) such that for each i, (Ui , T|Ui , O|Ui ) is isomorphic to an affinoid. A morphism X → Y between two rigid analytic spaces is a morphism between the associated locally ringed G-spaces. ...
... Definition 1.15. A rigid analytic space is a locally ringed G-space (X, T , O) admitting a covering {Ui } ∈ Cov(X) such that for each i, (Ui , T|Ui , O|Ui ) is isomorphic to an affinoid. A morphism X → Y between two rigid analytic spaces is a morphism between the associated locally ringed G-spaces. ...
Algebraic Groups I. Homework 10 1. Let G be a smooth connected
... (iii) Working over k and using suitable left and right translations by geometric points, prove that dµ(ξ) is an isomorphism for all k-points ξ of UG (λ−1 ) × PG (λ). Deduce that if UG (λ−1 ) and PG (λ) are smooth (OK for GL(V ) by (ii)) then µ induces an isomorphism between complete local rings at a ...
... (iii) Working over k and using suitable left and right translations by geometric points, prove that dµ(ξ) is an isomorphism for all k-points ξ of UG (λ−1 ) × PG (λ). Deduce that if UG (λ−1 ) and PG (λ) are smooth (OK for GL(V ) by (ii)) then µ induces an isomorphism between complete local rings at a ...
NOTES hist geometry
... 7. There exist two triangles that are similar but not congruent. 8. There exists a pair of lines that are everywhere equidistant. 9. Given any three non-collinear points, there is a circle that passes through them. 10. Given any area, there is a triangle whose area is greater than the given area. To ...
... 7. There exist two triangles that are similar but not congruent. 8. There exists a pair of lines that are everywhere equidistant. 9. Given any three non-collinear points, there is a circle that passes through them. 10. Given any area, there is a triangle whose area is greater than the given area. To ...