For this introduction we will be focusing only on type \(A\), and the main object of study is the symmetric group \(S_n\). \(S_n\) for our purposes will be the group of all bijective functions from the set \([n]=\{1,2,\ldots,n\}\) to itself. There are natural injective homomorphisms \(S_n\hookrightarrow S_{n+1}\) for all \(n\) obtained by considering \([n]\) as a subset of \([n+1]\); the image of a function in this homomorphism \(S_n\hookrightarrow S_{n+1}\) will fix \(n+1\) and do to the rest of the elements what it is doing already. It will be convenient to consider all of these groups to be sitting inside \(S_\infty\), which is the group of bijective functions \(f:\mathbb N\to\mathbb{N}\) such that \(f(i)=i\) for all but finitely many \(i\). Essentially, $$S_\infty=\bigcup_{n=1}^{\infty}{S_n}$$
Let \(F\) be a field of characteristic zero. We turn to the polynomial ring \(F[x_1,x_2,\ldots]\) in countably many indeterminates, which we will for convenience refer to as \(R\). \(S_\infty\) acts on \(R\) by \(F\)-automorphisms (that is, automorphisms that are \(F\)-linear). Specifically, for \(f\in S_\infty\) the action of \(f\) on \(x_i\) is $$f\cdot x_i=x_{f(i)}$$ There is only one way to extend this to an \(F\)-automorphism of \(R\). Essentially \(f\)'s action is the substitution \(x_i\mapsto x_{f(i)}\) in the usual sense. This action, while natural, is not commonly used in the literature because while it works perfectly well as a left action on the polynomials themselves, meaning \(f\cdot (g\cdot p)=(f\circ g)\cdot p\), it does not come from a left action on the variables. If you move the action inside the parentheses of the polynomial \(p(x_1,\ldots)\), viewed as a function, the order of multiplication in \(S_\infty\) is reversed. Despite this, this will be the most convenient action for our purposes.
Now we will need to refer to specific elements of \(S_\infty\), namely the adjacent transpositions \(s_i\) defined by $$s_i(j)=\left\{\begin{array}{cc}i+1&\mbox{ if }j=i\\i&\mbox{ if }j=i+1\\j&\mbox{ otherwise}\end{array}\right.$$ The elements \(s_i\) generate \(S_\infty\) as a Coxeter group, which is very important and will be discussed later, but we will not dwell on it now. Now we are interested in defining the divided difference operators \(\partial^i\) for \(i\in\mathbb{N}\). \(\partial^i\) takes elements of \(R\) to elements of \(R\) and is \(F\)-linear, and is defined abstractly as follows: $$\partial^i=\frac1{x_i-x_{i+1}}(1-s_i)$$ If this definition is too abstract for you, that's no problem. It is sufficient to know that \(\partial^i\) acts as follows: $$\partial^i(p)=\frac{p-s_i\cdot p}{x_i-x_{i+1}}$$ That is, we take the polynomial, switch two of the variables, subtract from the original, and divide by the difference of the two variables we swapped. It is not obvious that this always yields a polynomial, and you may want to convince yourself of that before moving on.
Now we define the operators \(\partial^w\) for all \(w\in S_\infty\) built from the \(\partial^i\). Namely, if \(w\) can be expressed as a specific product of adjacent transpositions \(w=s_{i_1}\cdots s_{i_k}\) with \(k\) as small as possible (so the expression is as short as possible) then we define $$\partial^w=\partial^{i_1}\cdots\partial^{i_k}$$ If we take such a product where the expression is not as short as possible, we instead get \(0\).
It is not difficult to work out a product rule for \(\partial^i\). You may want to try it yourself, but in any case the formula is $$\partial^i(pq)=\partial^i(p)q+p\partial^i(q)+(x_{i+1}-x_{i})\partial^i(p)\partial^i(q)$$ We can iterate this to obtain a product rule for \(\partial^w\) for arbitrary permutations \(w\). We end up with a formula of the form $$\partial^w(pq)=\sum_{u,v\in S_\infty}{c_{u,v}^w\partial^u(p)\partial^v(q)}$$ where the coefficients \(c_{u,v}^w\) are polynomials depending on \(u\), \(v\), and \(w\).
You may be surprised to know that by now I have already made good on my promise of bringing you from zero to Schubert calculus in minutes. The coefficients \(c_{u,v}^w\), the equivariant Littlewood-Richardson coefficients, are one of the central objects of study in Schubert calculus, and finding a formula of a certain type for them is a hopelessly difficult unsolved problem. While irrelevant to our geometry-free investigation, it is interesting to note that the polynomials \(c_{u,v}^w\) are the structure constants in the equivariant cohomology rings of complete flag varieties over the complex numbers (rings we will construct combinatorially later), and whenever \(c_{u,v}^w\) is an integer it counts the number of points in transverse triple intersections of Schubert varieties. It is known via geometric proofs (see Graham, Duke Math. J. Volume 109, Number 3 (2001), 599-614) that \(c_{u,v}^w\) is a polynomial in the differences \(x_{i+1}-x_i\) with nonnegative integer coefficients. This has not yet been paralleled combinatorially except in special cases, but we're working on it.
In the next post we will build our general framework for equivariant Schubert calculus.
Very well said. Keep up the good work. Glad you are exploring this.
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