1 results
Quasisymmetric harmonics of the exterior algebra
- Part of
-
- Journal:
- Canadian Mathematical Bulletin / Volume 66 / Issue 3 / September 2023
- Published online by Cambridge University Press:
- 10 January 2023, pp. 997-1013
- Print publication:
- September 2023
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let
$R_n$ denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in
$R_n$ form a commutative subalgebra of 
$R_n$.(2) There is a basis of the quotient of
$R_n$ by the ideal 
$I_n$ generated by the quasisymmetric polynomials in 
$R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by where
$$ \begin{align*}\text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,\end{align*} $$
$f^{(n-k,k)}$ is the number of standard tableaux of shape 
$(n-k,k)$.(3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.
