Sandpile Groups of Cayley Graphs over 𝔽₂ʳ: Sylow-2 Structure, Isospectral Counterexamples, and Boolean Incidence Algebra

Abstract

This paper determines new 2-primary sandpile invariants for Cayley multigraphs of the elementary abelian 2-group 𝔽₂ʳ and develops the Boolean incidence-algebra machinery underlying their computation. For the n-dimensional hypercube Qₙ, the paper determines the first previously unknown Sylow-2 cyclic exponent and thereby closes Conjecture 4.14 of Gao–Marx-Kuo–McDonald–Yuen. The exponent is given explicitly by cₙ(Qₙ) = max{ max₁≤ₓ<ₙ₋₁ (x + ν₂(x)), n − 3 + ν₂(n − 1) }. For an arbitrary connected Cayley multigraph of 𝔽₂ʳ, an exact parity theorem is proved for the number d(M) of Sylow-2 cyclic factors: d(M) is even ⇔ every nonzero generator has odd multiplicity. In this unique parity class, d(M) = 2ʳ − 2; for every other multiplicity-parity pattern, d(M) is odd. This replaces the previously proposed eigenvalue-valuation criterion by a complete classification in terms of the generator multiplicities modulo 2. The paper then constructs explicit isospectral counterexamples. For every r ≥ 3 there are primitive equal-degree Cayley multigraph pairs with identical complete Laplacian eigenvalue multisets but nonisomorphic sandpile groups. Their Sylow-2 factor counts are respectively 3·2ʳ⁻² − 1 and 2ʳ − 2. These families disprove both the published conjecture that the sandpile group is determined by the Laplacian eigenvalue set and the stronger earlier formulation asserting dependence only on the unlabeled eigenvalue multiset. A separate fixed-degree construction in 𝔽₂³ gives two generic 7-regular isospectral Cayley multigraphs with the same sandpile group ℤ/12ℤ ⊕ ℤ/60ℤ ⊕ ℤ/240ℤ but lying in distinct GL₃(𝔽₂)-orbits. Exact orbit enumeration proves that degree 7 is minimal for a same-degree critical-group collision among connected Cayley multigraphs of 𝔽₂³. The paper also derives the complete Smith normal form for every coprime weighted square with generator multiplicities (a,b,0): SNF(L♯ₐ,ᵦ) = diag(1,1,2ab(a+b)), and hence K(Cay(𝔽₂²;(a,b,0))) ≅ ℤ/(2ab(a+b))ℤ. The proof calculus is developed through Boolean incidence algebra. Indicator vectors identify set operations with exact polynomial and matrix operations; the Boolean zeta and Möbius transforms receive recursive Kronecker factorizations; inclusion–exclusion and exact-membership inversion are expressed as matrix transforms; and staged Boolean-lattice transforms are given exact operation counts. The paper also proves a deterministic worst-case optimal Θ(nk) algorithm for computing the cardinality of the union of k explicitly represented subsets of an n-element universe. Further extensions treat bounded-multiset product posets, product-of-chains zeta and Möbius transforms, spectral formulas, sparse Möbius-transform questions, and quantum realizations of set operations. The resulting framework connects sandpile groups, Smith normal forms, Boolean group algebras, incidence algebras, chip-firing, structured transforms, and exact combinatorial computation. 20 Keywords