DECOMPOSITION NUMBERS FOR WEIGHT THREE BLOCKS OF

... Hn (we shall abuse notation by saying that λ and µ lie in this block) if and only if they have the same e-core; this means that they also have the same weight, and this is called the (e-)weight of the block. If there is a bead in an abacus display for λ with exactly w empty spaces above it on the sa ...

... Hn (we shall abuse notation by saying that λ and µ lie in this block) if and only if they have the same e-core; this means that they also have the same weight, and this is called the (e-)weight of the block. If there is a bead in an abacus display for λ with exactly w empty spaces above it on the sa ...

Hovhannes Khudaverdian's notes

... Examples. Q, R, C are fields. Z is not a field, Space M at[p × p] of p × p matrices is not a field too. Another important example: Let (R, +, ×) be an integral domain. Then one can consider its field of fractions: pairs f /g with g 6= 0 and indentification f /g = f 0 /g 0 if f g 0 = f 0 g. In this w ...

... Examples. Q, R, C are fields. Z is not a field, Space M at[p × p] of p × p matrices is not a field too. Another important example: Let (R, +, ×) be an integral domain. Then one can consider its field of fractions: pairs f /g with g 6= 0 and indentification f /g = f 0 /g 0 if f g 0 = f 0 g. In this w ...

Introduction to Lie Groups

... theory of Lie groups. Students are expected to a have an undergraduate level background in group theory, ring theory and analysis. We focus on the so-called matrix Lie groups since this allows us to cover the most common examples of Lie groups in the most direct manner and with the minimum amount of ...

... theory of Lie groups. Students are expected to a have an undergraduate level background in group theory, ring theory and analysis. We focus on the so-called matrix Lie groups since this allows us to cover the most common examples of Lie groups in the most direct manner and with the minimum amount of ...

Mathematics of Cryptography

... This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter has several objectives: ❏ To review integer arithmetic, concentrating on divisibility and finding the greatest common divisor using the Euclidean algorithm ❏ To understand how the extended Euclidean ...

... This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter has several objectives: ❏ To review integer arithmetic, concentrating on divisibility and finding the greatest common divisor using the Euclidean algorithm ❏ To understand how the extended Euclidean ...

Lie Algebras and Representation Theory

... As an exercise in that direction, the reader should try to describe the orbits of the action of GL(n, R) on Rn × Rn defined by A · (x, y) := (Ax, Ay). 1.2. Group representations. A representation of a group G is simply an action of G on a K–vector space V by linear maps. So we need an action ϕ : G × ...

... As an exercise in that direction, the reader should try to describe the orbits of the action of GL(n, R) on Rn × Rn defined by A · (x, y) := (Ax, Ay). 1.2. Group representations. A representation of a group G is simply an action of G on a K–vector space V by linear maps. So we need an action ϕ : G × ...

# Jordan normal form

In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.