Krawtchouk polynomials, the Lie algebra $\mathfrak{sl}_2$, and Leonard pairs

Details Krawtchouk polynomials, the Lie algebra $\mathfrak{sl}_2$, and Leonard pairs Krawtchouk polynomials, the Lie algebra $\mathfrak{sl}_2$, and Leonard pairs

2012-01-08

arxiv.org/abs/1201.1645v1

Mathematics

Representation Theory

37 pages

Scientific paper

A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. In the present paper we give an elementary but comprehensive account of how the following are related: (i) Krawtchouk polynomials; (ii) finite-dimensional irreducible modules for the Lie algebra ${\mathfrak{sl}_2}$; (iii) a class of Leonard pairs said to have Krawtchouk type. Along the way we obtain elementary proofs of some well-known facts about Krawtchouk polynomials, such as the three-term recurrence, the orthogonality, the difference equation, and the generating function. The paper is a tutorial meant for a graduate student or a researcher unfamiliar with the above topics.

Affiliated with

Also associated with

No associations

Rating

Krawtchouk polynomials, the Lie algebra $\mathfrak{sl}_2$, and Leonard pairs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Krawtchouk polynomials, the Lie algebra $\mathfrak{sl}_2$, and Leonard pairs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Krawtchouk polynomials, the Lie algebra $\mathfrak{sl}_2$, and Leonard pairs will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-406