# Basis parametrized by the symmetric group elements for the coinvariant algebra

Let $$A_n$$ be the coinvariant algebra of the symmetric group $$S_n$$. This algebra has vector space dimension $$n!$$. $$A_n$$ is the quotient algebra of the polynomial ring $$K[x_1,...,x_n]$$ by the elementary symmetric polynomials $$e_i$$. Since $$e_1=x_1+....+x_n$$, one can "eliminate" the variable $$x_n$$ to obtain an algebra with admissible relations.

Question: Is there a canonical monomial basis $$b_p$$ (in the literature) for $$A_n$$ parametrised by the elements of the symmetric group $$p \in S_n$$ such that the variable $$x_n$$ does not appear?

There is the Artin basis, see theorem 1.7 in https://www.jpswanson.org/talks/2017_coinvariant_zoo.pdf ,which can do this up to a choice of a bijection to the symmetric group elements and I wonder what bijection to the symmetric group elements one should use (probably there is a convention or a canonical way for this already).

One idea might be to "set" $$r_i:=(i,i+1,...,n)$$, then the symmetric group should be having also the elements $$r_i$$ as a basis meaning all elements of the symmetric group have the unique form $$\Pi r_i^{a_i}$$ as in the Artin basis. Is this the canonical way that should be used?

• Lehmer code is one way, but there are many bijections from so-called 'sub-excedant functions' and permutations. Commented Aug 15, 2023 at 19:54

So the Artin basis already does this. There is a natural bijection of the symmetric group $$S_n$$ with sequences $$(a_1,\ldots,a_{n-1})$$ of integers $$a_i$$ with $$0\leq a_i\leq n-i$$. Namely, let $$w:[n]\to [n]$$ be a permutation. Then define $$a_i(w)=\#\{j>i\mid w(i)>w(j)\}$$ This is known as the Lehmer code. It's a little bit more work to get back to the permutation from the Lehmer code, but it's still not very hard. For an algorithm, start with a $$0$$-indexed sequence $$A^0=(1,2,\ldots,n)$$. To construct $$w$$ given $$(a_1,\ldots,a_{n-1})$$, at iteration $$i$$ set $$w(i)=A^{i-1}_{a_i}$$ and pop'' the element at index $$a_i$$ to obtain a new $$0$$-indexed sequence $$A^i$$. For the last index $$w(n)$$, the value will be the only remaining element in $$A^{n-1}$$.
If you just want polynomials and not monomials, then a very important basis is the Schubert polynomials. These are defined by the initial condition $$S_{w_0} = x_1^{n-1} x_2^{n-2} \cdots x_{n-1}$$ and the recursive relation: If $$w(i) > w(i+1)$$, then $$S_{w s_i} = \frac{S_w(x_1, x_2, \ldots, x_i, x_{i+1}, \ldots, x_n)-S_w(x_1, x_2, \ldots, x_{i+1}, x_i,\ldots, x_n)}{x_i - x_{i+1}}.$$
It is not obvious that these recursive relations are all compatible, but they are. The vector space spanned by all Schubert polynomials is the same as the vector space spanned by the Artin monomials, with the leading term of $$S_w(x_1, \ldots, x_n)$$ being the Artin monomial assigned to $$w$$ in Matt Samuel's answer.