I introduce an effort to catalog the gauge-invariant interactions of Standard Model (SM) particles and new fields in a variety of representations of the SM color gauge group. In this first installment, we direct this effort toward fields in the six-dimensional (sextet) representation. I consider effective operators of mass dimension up to seven (comprehensively up to dimension six), featuring both scalar and fermionic color sextets. I use an iterative tensor-product method to identify the color invariants underpinning such operators, emphasizing structures that have received little attention to date. In order to demonstrate the utility of our approach, I study a simple but novel model of color-sextet fields at the Large Hadron Collider (LHC). I compute cross sections for an array of new production channels enabled by our operators, including single-sextet production and sextet production in association with photons or leptons. I also discuss dijet-resonance constraints on a sextet fermion. This example shows that there remains a wide array of fairly minimal but well motivated and unexplored models with extended strong sectors as we await the high-luminosity LHC.
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