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Definitions of Symbols used in this Website 
An excellent general reference (and essentially the bible for research in this area) for symmetric functions is the book Symmetric Functions and Hall Polynomials by I.G. Macdonald, Oxford Science Publications, Second Edition 1995. There are other references for symmetric functions, Segan
are the Schur symmetric functions 
are the homogeneous symmetric functions 
are the power symmetric functions 
are the elementary symmetric functions 
are the monomial symmetric functions 
Define an inner product
by the values on the power symmetric functions by .
This is the standard inner product on symmetric functions.
The Schur symmetric functions have the property that they are upper triangularly related to the monomial symmetric functions and orthogonal with respect to the inner product. 
Define an inner product
by the values on the power symmetric functions by .
This is usually referred to as the HallLittlewood scalar product.
Define the symmetric functions to be the symmetric functions with the property . These are the dualSchur symmetric functions. 
Define another inner product
by the values on the power symmetric functions by
This is the Macadonald inner product or the q,tinner product. 
The Macdonald symmetric functions Pmu(X,q,t) are defined by the property
that they are upper triangularly related to the monomial symmetric functions
and orthogonal with respect to the
inner product. That is, they satisfy the following two conditions

Let
where
has the expansion sum where the sum is over all partitions of n and this is the defintion of the q,tKostka coefficients, . The are usually referred to as the integral form of the Macdonald symmetric functions. 
Define to be another class of symmetric functions. I find that these are slighly easier to work with than the . 
Define the basis as a transformed basis for the symmetric functions. This basis is important because it is conjectured to be the Frobenius characteristic of the n! conjecture module. The corresponding transformed coefficients will be defined as . 
The HallLittlewood symmetric functions that appear here are transformed from the standard symmetric functions. They are given by the formula where the first sum is over 1 ¾ i < j ¾ n and the second sum is over all j and 1 ¾ i ¾ n. The are analogous to the (in fact equal when q = 0) but are a transformation of the symmetric functions normally referred to as HallLittlewood polynomials. The transition coefficients with the Schur symmetric basis are the tKostka coefficients. 
Occasionally, plethystic notation is used for symmetric functions here.
This is a notational device for expressing the substitution of the monomials
of one expression, $E = E(t_1, t_2, t_3, \ldots)$ for the variables of
a symmetric function, $P$. The result will be denoted by $P[E]$ and represents
the expression found by expanding $P$ in terms of the power symmetric functions
and then substituting for $p_k$ the expression $E(t_1^k, t_2^k, t_3^k,
\ldots)$.
More precisely, if the power sum expansion of the symmetric function $P$ is given by $$P = \sum_\la c_\la p_\la$$ then the $P[E]$ is given by the formula $$P[E] = \sum_\la c_\la p_\la \coeff_{p_k \rightarrow E(t_1^k, t_2^k, t_3^k, \ldots)}$$. To express a symmetric function in a single set of variables $x_1, x_2, \ldots, x_n$, let $X_n = x_1 + x_2 + \cdots + x_n$. The expression $P[X_n]$ represents the symmetric function $P$ evaluated at the variables $x_1, x_2, \ldots, x_n$ since $$P(x_1, x_2, \ldots, x_n) = \sum_\la c_\la p_\la \coeff_{p_k \rightarrow x_1^k + x_2^k + \cdots + x_n^k} = P[X_n]$$ 