The multiplicative structure of the K-theoretical McKay correspondence for the Hilbert scheme of points in the complex plane

Technical Summary: The Multiplicative Structure of the K-Theoretical McKay Correspondence for the Hilbert Scheme of Points in the Complex Plane

Problem Statement
The paper investigates the K-theory of the Hilbert scheme of $n$ points in the complex plane, denoted $Hilb^n(\mathbb{A}^2_{\mathbb{C}})$. Under the K-theoretical McKay correspondence, the equivariant K-theory of this scheme, $K^{S_n}(\mathbb{C}^{2n})$, is isomorphic to the space of symmetric functions of degree $n$, $\Lambda_n$. The authors address two primary questions regarding the multiplicative structure induced on $\Lambda_n$ by the tensor product in the K-theory ring $K(Hilb^n)$:

1. The Endomorphism Problem (A): For a given class $F \in K(Hilb^n)$, determine a compact formula for the induced endomorphism $E_F: \Lambda_n \to \Lambda_n$ such that the diagram involving the McKay isomorphism $\Theta$ and tensor multiplication by $F$ commutes. Specifically, the authors focus on $F$ being the Adams powers $\psi^k B_n$ of the tautological vector bundle $B_n$, which generate the K-theory ring.
2. The Structure Constants Problem (B): Describe the structure constants $c^\mu_{\lambda_1, \lambda_2}$ for the multiplication $\odot$ on $\Lambda_n$ induced by the tensor product, specifically in the power sum basis.

Methodology
The authors employ a combination of geometric intuition and combinatorial reduction.

• Geometric Reduction: While the standard torus action $T = (\mathbb{C}^*)^2$ on the Hilbert scheme yields a finite fixed point set, the combinatorics of tangent weights are complex. The authors introduce a reduction to a one-dimensional coordinate subtorus $T_q \subset T$. Under this restriction (corresponding to the substitution $t=1$ in combinatorial formulas), the fixed point set $Hilb^{T_q}_n$ becomes a disjoint union of affine spaces. This simplifies the geometry of the fixed point set and, crucially, simplifies the combinatorics of the tangent weights.
• Combinatorial Analysis: The proof relies on the properties of Macdonald polynomials $\tilde{H}_\lambda(q,t)$ and their restriction to $t=1$. The authors utilize the Frobenius characteristic map to relate the K-theory of the Hilbert scheme to symmetric functions. They analyze the operators $\nabla_F$ (adjoints of the multiplication operators) and demonstrate that under the restriction to $T_q$, these operators satisfy the Leibniz rule, a property not present in the full equivariant setting.
• Localization and Eigenbases: Using the localization theorem, the authors work with the basis of fundamental classes of fixed points. They establish that the modified Macdonald polynomials (restricted to $t=1$) form an eigenbasis for the relevant operators, allowing for explicit computation of the action of Adams powers.

Key Contributions and Results

1. Proof of Boissière's Conjecture: The primary result is the proof of a formula conjectured by Boissière [Boi06b] for the endomorphism $E_{\psi^k B_n}$. The authors establish that for any natural number $k$:
   $$E_{\psi^k B} = \sum_{\lambda} \frac{k^{l(\lambda)-1} s(\lambda)}{\prod_{i \in \mathbb{N}} a_i(\lambda)!} p_{s(\lambda)} \partial_\lambda$$
   where the sum runs over partitions $\lambda$, $l(\lambda)$ is the length, $s(\lambda)$ is the sum, $a_i(\lambda)$ is the multiplicity of part $i$, and $\partial_\lambda$ is the differential operator associated with the partition. This is presented as Theorem 5.15. An equivalent formulation involving the operator $\nabla^*_k$ is also provided.

2. Leibniz Rule via Subtorus Restriction: A central technical contribution is the demonstration that the restriction of the operator $\nabla_k$ to the subtorus $T_q$ (setting $t=1$) satisfies the Leibniz rule: $\nabla^q_k(fg) = \nabla^q_k(f)g + f\nabla^q_k(g)$. This property allows the authors to derive closed formulas for the operators by computing their action on the generators (power sums) and extending via the product rule.

3. Structure Constants for Tensor Product: In Section 7, the authors derive a formula for the structure constants $c^\mu_{\lambda_1, \lambda_2}$ of the multiplication $\odot$ on $\Lambda_n$. Theorem 7.3 expresses these constants as coefficients of a specific power series involving the inner product of symmetric functions and a weight function $W_\nu(q)$. The formula is given by:
   $$c^\mu_{\lambda_1, \lambda_2} = \text{Coeff}_{q-1} \left( l(\lambda_1) + l(\lambda_2) - n - l(\mu); \frac{[\mu]_q}{z_\mu [\lambda_1]_q [\lambda_2]_q} \cdot \sum_{\nu \vdash n} \frac{\langle p_{\lambda_1}, h_\nu \rangle \langle p_{\lambda_2}, h_\nu \rangle \langle p_\mu, m_\nu \rangle}{W_\nu(q)} \right)$$
   This result clarifies the vanishing conditions for these constants based on the degrees of the partitions.

4. Correction of Previous Literature: The authors note a discrepancy in the computational results of [Boi06b] regarding the product $p_{22} \odot p_{22}$. Using their derived formula and dimensional arguments (Remark 7.5), they assert that the product should be zero, suggesting a misprint in the earlier work.

Significance and Claims
The paper claims to resolve a specific conjecture regarding the K-theoretical McKay correspondence, providing a closed-form expression for the action of Adams powers of the tautological bundle on symmetric functions. The authors emphasize that their proof is "purely combinatorial" but is grounded in the geometric intuition of restricting the torus action to a subtorus to simplify the fixed point geometry.

The significance lies in:

• Providing the first proof of Boissière's conjecture for arbitrary $k$.
• Establishing the multiplicative structure constants for the K-theory induced product on symmetric functions, a structure distinct from the cohomological case due to the non-graded nature of K-theory.
• Demonstrating that the restriction to a subtorus $T_q$ is a powerful tool for simplifying complex equivariant K-theoretic calculations, turning them into tractable combinatorial problems involving the Leibniz rule.

The authors do not claim to extend these results to other varieties or groups within this work, nor do they propose new experimental applications. The work is presented as a definitive solution to the specific algebraic and combinatorial problems posed by the K-theoretical McKay correspondence for the Hilbert scheme of points in the plane.