The Symmetric Hit Problem in Four Variables: Relative Cohits and Stabilizer-Fixed Preimages

Abstract

In this work, we study the symmetric hit problem in four variables by analyzing the relative cohit space, defined as the kernel of the inclusion-induced map from symmetric to ordinary cohits. Theoretically, we establish that this relative space is isomorphic to a kernel in the first group cohomology of the permutation group and, equivalently, to an ${\mathrm{Ext}}^1$ kernel for the general linear group, equipped with a canonical action of the normalizer quotient. Furthermore, bypassing conditional descent hypotheses, we utilize explicit Lucas--Cartan identities to construct unconditional fixed Steenrod preimages for nontrivial stabilizers, resolving infinite families of monomials strictly above the minimal-spike weight. Using the ordinary and symmetric Kameko isomorphisms, we reduce the determination of complete bases in two generic degree families to exact calculations in degrees one, three, six, ten, and sixteen. We discover that universal relative vanishing fails: the relative space is non-trivially one-dimensional in degree fifteen and forms an indecomposable two-dimensional normalizer module in degree eighteen. In the family $5\cdot 2^t-4$, the symmetric cohit dimension stabilizes at six and the relative dimension at one for $t\ge 2$; in the family $7\cdot 2^t-4$, they are four and zero for $t\ge 1$. We give complete bases by explicit exponent formulas and prove Janfada's monomial implication in every degree of both families. A general numerical threshold determines the finite initial degree from which Kameko isomorphisms apply, and a direct-sum formula transports the entire symmetric hit basis.

arXiv comment36 pages. Comments are welcome!