Symmetric functions
Algebraic Combinatorics February 22, 2007
The monomial symmetric functions are denoted in ACE by m[λ]. If we want to work with variables rather than bases, the ACE function SfEval does this conversion for us. For example, SfEval(m[1, 1, 1], x[1] + x[2] + x[3] + x[4]); will output the monomial expansion of the monomial symmetric function m[1, 1, 1] in four variables. Note, the Maple functions factor,simplify, expand modify the appearance of your polynomials.
Exercise 1: Use ACE to show that your answer to problem 1 in the homework is correct.
The function in ACE to expand a symmetric function (expressed in terms of a basis) into the monomial basis is Tom.
Exercise 2: Use ACE to verify your answer to problem number 3.
Exercise 3: Practice on other examples to convince yourself that the only monomial symmetric functions that occur in m1k mλ are those indexed by λ plus a vertical strip.
The ACE function Sf2Table(symfun,’b’) takes a symmetric function in some basis b and computes a table associating the coefficient of bi to i. Experiment with this function. In particular, use the Maple help to find out how to use indices and entries.
Exercise 4: Use what you know about the product of m1k mλ to write a pro- gram that gives the set of partitions λ plus a vertical k-strip.
The function Tom only works if your polynomial is already in terms of a basis. So that we can use this function for any symmetric function,
Exercise 5: Write a program that takes a homogeneous polynomial in n vari- ables of degree k and outputs the expansion of this polynomial in terms of the monomial basis. Useful commands are coeff and ListPart.
1 Exercise 6: Now write a program that takes any symmetric function in n variables and returns the expansion in terms of the monomial basis.
Exercise 7: Check your answer to number 2 in the homework.
2 Schur Functions
We defined a Schur function by
X wt(T ) sλ(x1, . . . , xn) = x .
T ∈Tn(λ)
Recall that ACE has built in functions to compute the set of tableaux of a given shape λ and weight α. The function ListCompo has an option to allow zeros, refer to the help section.
Exercise 1. Write a program that takes a partition and an integer and returns the Schur function sλ(x1, . . . , xn).
Actually ACE also has built in functions to work with Schur functions as with the monomial basis. Use SfEval to check that your program works.
We can use the command T om(s[λ]) to expand a Schur function in terms of the monomial basis. Try this on some examples and note that the expansion is unitriangular.
Exercise 3: Recall your program that computes the number of tableaux of a given shape λ and weight µ. Write a program that compares the coefficient of the monomial term mµ in a Schur function sλ to this number.
Given that the monomial expansion of a Schur function is unitriangular, we know that the Schur functions form a basis for the symmetric function space. Therefore, as with T om, we can use T os to expand symmetric functions in terms of the Schur function basis.
Exercise 4: Examine the Schur function expansion of s`sλ. Determine the coefficients in this expansion. Determine precisely which Schur functions appear in this expansion? Repeat this exercise for the Schur function expansion of s1` sλ.
Exercise 5: Examine the Schur function expansion of sµsλ. Can you say anything about the coefficients or what terms occur in this expansion?
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