Shuffle formula in science fiction for Macdonald polynomials

We study the Macdonald intersection polynomials I_μ ⁽¹⁾, ,μ ⁽ᵏ⁾[X;q,t] , which are indexed by k -tuples of partitions μ ⁽¹⁾, ,μ ⁽ᵏ⁾ . These polynomials are conjectured to be equal to the bigraded Frobenius characteristic of the intersection of Garsia–Haiman modules, as proposed by the science fiction conjecture of Bergeron and Garsia. In this work, we establish the vanishing identity and the shape independence of the Macdonald intersection polynomials. Additionally, we unveil a remarkable connection between I_μ ⁽¹⁾, ,μ ⁽ᵏ⁾ and the character ∇ eₖ₋₁ of diagonal coinvariant algebra by employing the plethystic formula for the Macdonald polynomials of Garsia, Haiman, and Tesler. Furthermore, we establish a connection between I_μ ⁽¹⁾, ,μ ⁽ᵏ⁾ and the shuffle formula Dₖ₋₁[X;q,t] , utilizing novel combinatorial tools such as the column exchange rule and the lightning bolt formula for Macdonald intersection polynomials. Notably, our findings provide a new proof for the shuffle theorem.