UC San Diego

In this dissertation, I give a variety of combinatorial expansions for certain symmetric functions related to the modified Macdonald basis $\{\Ht_\mu\}_\mu$. We first prove the Delta Conjecture at $q=1$, giving an elementary basis expansion for $\Delta_{e_k} e_n \Big|_{q=1}$, where $\Delta_F$ is the eigenoperator for the modified Macdonald basis defined by setting $\Delta_F \Ht_\mu = F[B_\mu] \Ht_\mu$.

The main strategy is to expand the symmetric function as a generating series for an infinite, signed and weighted set of objects that give the elementary basis expansion (or $e$-expansion). To reduce this infinite sum to a finite polynomial with positive integer coefficients, we find a weight-preserving, sign-reversing involution whose fixed points all have positive sign. We then find a bijection to the conjectured combinatorial side of the Delta Conjecture at $q=1$.

This strategy leads us to give a combinatorial $e$-expansion of $\Delta_{m_\lambda} e_n \Big|_{q=1}$ in terms of (parallelogram) polyominoes, again by using a weight-preserving, sign-reversing involution. We will show that the coefficient of $e_\mu$ in this symmetric function $t$-counts (by area) the number of polyominoes whose bottom path has horizontal segments rearranging to $\lambda$ and whose top path has vertical segments rearranging to $\mu$. This result can be used to get combinatorial expansions for $\Delta_F e_\mu\Big|_{q=1}$, where $F$ is any of the classical bases $m_\lambda, f_\lambda, h_\lambda, e_\lambda, s_\lambda,$ or $p_\lambda$. This result uses the principal evaluation for the forgotten basis giving a new application for the forgotten symmetric functions.

The last result of this kind will give an $e$-expansion of $\Delta_{m_\lambda} (-1)^{n-1}p_n \Big|_{q=1}$ in terms of polyominoes whose top path ends in two East steps. The only difference is that now every polyomino appears with multiplicity the length of its return, defined as the first row (after the first vertical segment of the top path) at which the top and bottom path are one unit away. We once again get formulas for $\Delta_F (-1)^{n-1} p_n\Big|_{q=1}$, where $F$ is any of the classical bases. In general, $\Delta_{m_\lambda} \omega p_\mu\Big|_{q=1}$ is not $e$-positive, but we still present a formula in terms of power and elementary symmetric functions.

In the end, we will look at certain $e$-positivity phenomena and we conjecture that $\Delta_{m_\lambda} e_n$ and $\Delta_{ m_\lambda} (-1)^{n-1} p_n$, when expanded in terms of the elementary symmetric function basis, has coefficients which are polynomials in $u=q-1$ and $t$ with positive integer coefficients.

We will prove that the plethystic operators $B_a$ satisfy this phenomenon. In turn, we get a homogeneous basis expansion for the Hall-Littlewood polynomials and an elementary basis expansion for $\Delta_{e_k} e_n \Big|_{t=0}$, where the coefficients are polynomials in $u = q-1$ and $q$ with positive integer coefficients.