Let R be the Z-module generated by the irreducible characters of the symmetric group Sd . We determine bases for the kernel of the decomposition map. It is known that R ⊗Z F is isomorphic to the radical quotient of the Solomon descent algebra when F is a field of characteristic zero. We show that when F has prime characteristic and I d br is the kernel of the decomposition map for prime p then ...

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i) and (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal. (ii) There exists a basis for V with respect to whi...

An analytic hypersurface of M is a subset V ⊂ M such that for each point x ∈ V there exists an open set Ux ⊂ M containing x and a holomorphic function fx defined on Ux such that V ⊂ Ux is the zero-set of fx. Such an fx is called a local defining function for V near x. The quotient of any two local defining functions around x is a non-vanishing holomorphic function around x. An analytic hypersur...

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on c...

The notion of locally finite part of the dual coalgebra of certain quantized coordinate rings is introduced. In the case of irreducible flag manifolds this locally finite part is shown to coincide with a natural quotient coalgebra U of Uq(g). On the way the coradical filtration of U is determined. A graded version of the duality between U and the quantized coordinate ring is established. This l...

We calculate the fundamental groups π = π1(P rB) for all irreducible plane sextics B ⊂ P2 with simple singularities for which π is known to admit a dihedral quotient D10. All groups found are shown to be finite, two of them being of large order: 960 and 21600.

In this paper, we define a Hopf algebra structure on the vector space spanned by packed words using a selection-quotient coproduct. We show that this algebra is free on its irreducible packed words. Finally, we give some brief explanations on the Maple codes we have used.

The quantum plane is the non-commutative polynomial algebra in variables x and y with = q . In this paper, we study module variety of n -dimensional modules over plane, provide an explicit description its irreducible components their dimensions. We also describe dimensions GIT quotient respect to conjugation action GL

For an odd positive integer n ≥ 5, assuming the truth of the abc conjecture, we show that for a positive proportion of pairs (a, b) of integers the trinomials of the form tn + at + b (a, b ∈ Z) are irreducible and their discriminants are squarefree.