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This article illustrates the dynamical concept of homomesy in three kinds of dynamical systems -combinatorial, piecewise-linear, and birational -and shows the relationship between these three settings. In particular, we show how the rowmotion and promotion operations of Striker and Williams [16] can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley [14], and then lifted to birational operations on the positive orthant in R |P | and indeed to a dense subset of C |P | . When the poset P is a product of a chain of length a and a chain of length b, these lifted operations have order a + b, and exhibit the homomesy phenomenon: the time-averages of various quantities are the same in all orbits. One important tool is a concrete realization of the conjugacy between rowmotion and promotion found by Striker and Williams; this recombination map allows us to use homomesy for promotion to deduce homomesy for rowmotion. We also show that Stanley's transfer map between the order polytope and the chain polytope arises as the tropicalization of an analogous map in the bilinear realm.

Purpose: The role of immune-oncologic mechanisms of racial disparities in prostate cancer remains understudied. Limited research exists to evaluate the molecular underpinnings of immune differences in African American men (AAM) and European American men (EAM) prostate tumor microenvironment (TME). Experimental Design: A total of 1,173 radiation-naïve radical prostatectomy samples with whole transcriptome data from the Decipher GRID registry were used. Transcriptomic expressions of 1,260 immune-specific genes were selected to assess immune-oncologic differences between AAM and EAM prostate tumors. Race-specific differential expression of genes was assessed using a rank test, and intergene correlational matrix and gene set enrichment was used for pathway analysis. Results: AAM prostate tumors have significant enrichment of major immune-oncologic pathways, including proinflammatory cytokines, IFNα, IFNγ, TNFα signaling, ILs, and epithelial–mesenchymal transition. AAM TME has higher total immune content score (ICSHIGH) compared with 0 (37.8% vs. 21.9%, P = 0.003). AAM tumors also have lower DNA damage repair and are genomically radiosensitive as compared with EAM. IFITM3 (IFN-inducible transmembrane protein 3) was one of the major proinflammatory genes overexpressed in AAM that predicted increased risk of biochemical recurrence selectively for AAM in both discovery [HRAAM = 2.30; 95% confidence interval (CI), 1.21–4.34; P = 0.01] and validation (HRAAM = 2.42; 95% CI, 1.52–3.86; P = 0.0001) but not in EAM. Conclusions: Prostate tumors of AAM manifest a unique immune repertoire and have significant enrichment of proinflammatory immune pathways that are associated with poorer outcomes. Observed immune-oncologic differences can aid in a genomically adaptive approach to treating prostate cancer in AAM.