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arXiv2026

Equivariant Cohomological Crepant Resolution Conjecture for ADE-orbifolds

Let G be a non-trivial finite subgroup of SL_2(C) and let p : X \to C^2/G be the minimal resolution. We prove the C^*-equivariant Cohomological Crepant Resolution Conjecture for the ADE orbifold [C^2/G]. More precisely, after specializing the quantum parameters at suitable roots of unity, we prove that the Bryan-Gholampour transformation induces an isomorphism between the C^*-equivariant quantum cohomology of X and the C^*-equivariant Chen-Ruan cohomology of [C^2/G]. Our proof is direct and is based on the McKay correspondence, ADE root systems, and character theory. Following the character-theoretic form of the Bryan-Gholampour change of variables, we use the McKay correspondence to diagonalize the transformation and to reduce the compatibility of the products to a uniform root-system identity. In type A, this reduction admits a discrete Fourier realization. In type D, the cyclic subgroup of the binary dihedral group leads instead to finite sine transforms, while the exceptional cases E_6,E_7,E_8 are treated by exact matrix computations over the corresponding cyclotomic fields. As a preliminary result, for a symplectic complex vector space V and a finite subgroup G of Sp(V), we give an explicit presentation of the C^*-equivariant Chen-Ruan product of [V/G] and identify the resulting algebra with the Rees algebra of the center of the group algebra with respect to the age filtration. In particular, the equivariant Chen-Ruan algebra interpolates between the center of the group algebra and its associated graded algebra, the latter recovering the ordinary Chen-Ruan cohomology.

math.AG↗

arXiv2022

Assign Experiment Variants at Scale in Online Controlled Experiments

Online controlled experiments (A/B tests) have become the gold standard for learning the impact of new product features in technology companies. Randomization enables the inference of causality from an A/B test. The randomized assignment maps end users to experiment buckets and balances user characteristics between the groups. Therefore, experiments can attribute any outcome differences between the experiment groups to the product feature under experiment. Technology companies run A/B tests at scale -- hundreds if not thousands of A/B tests concurrently, each with millions of users. The large scale poses unique challenges to randomization. First, the randomized assignment must be fast since the experiment service receives hundreds of thousands of queries per second. Second, the variant assignments must be independent between experiments. Third, the assignment must be consistent when users revisit or an experiment enrolls more users. We present a novel assignment algorithm and statistical tests to validate the randomized assignments. Our results demonstrate that not only is this algorithm computationally fast but also satisfies the statistical requirements -- unbiased and independent.

stat.AP↗