REU 2010

This page contains information regarding the Summer 2010 REU project under the guidance of Professor Siddhartha Sahi.

Update, as of 7/21/2010: I am currently investigating the case where L=M and N=(1,1).

Project Description

We shall study the normalized binomial coefficients (henceforth Bi) and the normalized Littlewood-Richardson coefficients (henceforth Lrc) of Jack polynomials. It is conjectured that both Bi and Lrc are always polynomials with positive integral coefficients, and a varying degree of constraints on the coefficients have already been proven. With the aid of the new computational algorithm for Bi and Lrc, we hope to gain an insight into the inner workings of these coefficients—possibly coming up with closed-form formulas for the coefficients in some cases.

Supplementary Resources

Updates

6/7/2010 I have spent most of the first week summoning a livable space within my new apartment; I certainly did not imagine this to be as time-consuming as it has been. In the meantime, I have met with Professor Sahi and discussed the project. See the project description for an abstract; a more detailed set of notes will be posted soon.

6/11/2010 I spent the week reading bits and pieces of a few textbooks, trying to have (at least) some understanding of the mathematical objects I will be playing around with. I have read enough to write a few pages of notes, but they are woefully incomplete as of now. I suppose I don’t really understand well what I’ll be dealing with. Meanwhile, I did my obligatory presentation at DIMACS at 10 today.

This is the first “assignment” from Prof. Sahi:

It is known that \mathrm{Lrc}(L,M,N)=0 unless

  1. L \supset M and L \supset N;
  2. |L| \leq |M|+|N|.

Hence, if we fix M and N, then figuring out all possible L‘s that produces nonzero \mathrm{Lrc} is a finite process. To make the matters simpler, we take N=(1^r), and consider the following questions—

  1. List all possible nonzero coefficients, given a fixed partition M and r \in \mathbb{N}.
  2. Given a fixed partition M, which L‘s produce nonzero coefficients?
  3. Is \mathrm{Lrc} always factorizable into linear factors?
  4. Can you conjecture a formula?

6/20/2010 After a meeting with Prof. Sahi on Friday and a weekend of digging into papers, I have finally understood the basic notions I am dealing with in this project! This is a little pathetic, to be sure, but at least I can work on the problems now. The notes explaining the said notions are up.

7/8/2010 I now have a clearer sense of direction, equipped with methods of investigation that I can actually wield competently; I expect to obtain some new closed-form formulas for lrc—however limited in scope they may be—in the near future. In the meantime, I have indulged myself in coming up with a trivial formula or two, and had some bowls of cereals. Oh, and a glass of watermelon juice.

A few more words on the current status of the project: as per Prof. Sahi’s recommendation, I am now focusing my attention on the so-called non-Pieri cases of the Littlewood-Richardson coefficients. A Pieri case satisfies the following:

  1. |L|=|M|+|N|;
  2. L \supset M and L \supset N;
  3. Considering L and M as Young diagrams, L is obtained by adding boxes to M; at most one box may be added to each row of M.

A non-Pieri case is then obtained by relaxing the condition 1 above:

  1. |L| \leq |M|+|N|.

There is an explicit formula for lrc in the Pieri case, described in the notes. The goal is to generalize the formula to non-Pieri cases.

7/21/2010 I am currently investigating the case where L=M and N=(1,1).