A 𝑞-deformation of an algebra of Klyachko, and Macdonald’s reduced word formula

There is a striking similarity between Macdonald’s reduced word formula and the image of the Schubert class in the cohomology ring of the permutahedral variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper P normal e normal r normal m Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">P</mml:mi> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathrm {Perm}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as computed by Klyachko. Toward understanding this better, we undertake an in-depth study of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -deformation of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {S}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant part of the rational cohomology ring of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper P normal e normal r normal m Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">P</mml:mi> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathrm {Perm}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which we call the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Klyachko algebra. We uncover intimate links between expansions in the basis of squarefree monomials in this algebra and various notions in algebraic combinatorics, thereby connecting seemingly unrelated results by finding a common ground to study them. Our main results are as follows. A <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -analogue of divided symmetrization ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -DS) using Yang–Baxter elements in the Hecke algebra. It is a linear form that picks up coefficients in the squarefree basis. A relation between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -DS and the ideal of quasisymmetric polynomials involving work of Aval–Bergeron–Bergeron. A family of polynomials in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with nonnegative integral coefficients that specialize to Postnikov’s mixed Eulerian numbers when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q equals 1"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">q=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We refer to these new polynomials as remixed Eulerian numbers. For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">q&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , their normalized versions occur as probabilities in the internal diffusion limited aggregation stochastic process. A lift of Macdonald’s reduced word identity in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Klyachko algebra. The Schubert expansion of the Chow class of the standard split Deligne–Lusztig variety in type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a prime power.