The behavior of Stanley depth under polarization

Let K be a field, R = K[X1, . . . ,Xn] be the polynomial ring and J ( I two monomial ideals in R. In this paper we show that sdepth I/J− depth I/J = sdepth I /J p − depth I /J , where sdepth I/J denotes the Stanley depth and I p denotes the polarization. This solves a conjecture by Herzog [Her13] and reduces the famous Stanley conjecture (for modules of the form I/J) to the squarefree case. As a consequence, the Stanley conjecture for algebras of the form R/I and the well-known combinatorial conjecture that every CohenMacaulay simplicial complex is partitionable are equivalent.