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(2) M AIN RESULT: TYPE B The type B Eulerian polynomial is: Bd (z) =. d X k=0. hk (⇡dB )z k =. X. z exc(⌧ )+b. neg(⌧ )+1 c 2. ⌧ 2S± d. Theorem (B- ’20) For any flat X 2 L[Bd ] and r = 0, 1, . . . , d,. n o neg(⌧ ) + 1 ⌘X (⇧r (⇡dB )) = # ⌧ 2 S± : fix(⌧ ) = X, exc(⌧ ) + b c = r . d 2.
(3) C OROLLARIES OF THE PROOFS The generating functions for Eulerian polynomials are (Euler 1700s, Brenti ’94): A(z, x) =. X. Ad (z). d 0. xd z 1 = d! z ex(z. B(z, x) =. 1). X d 0. Bd (z). xd (1 z)ex(1 = d 2 d! 1 zex(1. z)/2 z). The proofs of the previous theorems show: X⇣ X d 0. X⇣ X d 0. ⌧ 2S± d. tdim(fix(. )) exc( ). 2Sd. tdim(fix(⌧ )) z excB (⌧ ). z. ⌘ xd d!. = A(z, x)t. ⌘ xd t 1 = B(z, x) A(z, x) 2 d 2 d!. (Brenti ’00). (B- ’20).
(4) QUESTION (Posed by Ardila-Castillo-Eur-Postnikov ’20) What is a nice type B analog of the following result? Theorem (Postnikov ’09, Ardila-Benedetti-Doker ’10) Every generalized permutahedron in Rd can be written uniquely as a signed Minkowski sum of the faces of the standard simplex [d] . Cone of generalized permutahedra (elements of the form log[p] in ⇧1 (⇡3 )):.
(5) T YPE B ANALOG OF. d. The faces of the cross-polytope only generate a subspace of (roughly) half the required dimension. e3 e1. e2. 12. e2. e1 e3. =. 123. =. 12 12. +. 13. +. 23. 123. (up to translation).
(6) S ECOND COROLLARY. Corollary (B- ’20) Any family of generators (via signed Minkowski sums) for generalized permutahedra of type B contains at least 2d 1 full-dimensional polytopes. I From the previous theorem, ⌘{0} (⇧1 (⇡dB )) = 2d. 1.. I We employ Eulerian idempotents of R⌃[Bd ] (Saliola ’06, Aguiar-Mahajan ’17) to deduce that, if q is not full-dimensional, the projection of log[q] to certain ⌘{0} (⇧1 (⇡dB ))-dimensional subspace is zero. 14 full-dimensional type B shard polytopes in R3 (Padrol-Pilaud-Ritter ’20).
(7) G ENERATORS FOR TYPE B GENERALIZED PERMUTAHEDRA For nonempty S ✓ [±d] with S \ S = ; define S. = Conv{ej | j 2 S} 0 12. and. 0 S. = Conv(. S. [ {0}). (ei = ei ). e3. 0 12. e1. 12. e2. 2. Theorem (B- ’20) Any type B generalized permutahedron can be written uniquely as a signed Minkowski sum of the simplices S,. 0 S. : min{|j| : j 2 S} 2 S .. The proof uses a valuation ⇧1 (⇡dB ) ! R⌃[Bd ]..
(8) G ENERATORS FOR TYPE B GENERALIZED PERMUTAHEDRA For nonempty S ✓ [±d] with S \ S = ; define S. = Conv{ej | j 2 S} 0 12. and. 0 S. = Conv(. S. [ {0}). (ei = ei ). e3. 0 12. 12. e1. e2. 2. Theorem (B- ’20) Any type B generalized permutahedron can be written uniquely as a signed Minkowski sum of the simplices 0S . ". Change of. basis. ". bisobmodular functions. <→. signed Mink of. bog. -. sum.
(9) T HANK YOU !. arXiv:2009.05876.
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