Generalized quotients in Coxeter groups

Abstract

For (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) a Coxeter group, we study sets of the form <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W slash upper V equals StartSet w element-of upper W vertical-bar l left-parenthesis w v right-parenthesis equals l left-parenthesis w right-parenthesis plus l left-parenthesis v right-parenthesis for all v element-of upper V EndSet comma"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>V</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>w</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>w</mml:mi> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thickmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>for all</mml:mtext> </mml:mrow> <mml:mspace width="thickmathspace" /> <mml:mi>v</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>V</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">W/V = \{ w \in W|l(wv) = l(w) + l(v)\;{\text {for all}}\;v \in V\} ,</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V subset-of-or-equal-to upper W"> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mi>W</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">V \subseteq W</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Such sets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W slash upper V"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">W/V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, here called <italic>generalized quotients</italic>, are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V subset-of-or-equal-to upper S"> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">V \subseteq S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (i.e., for minimal coset representatives modulo a parabolic subgroup). We show that Bruhat intervals in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W slash upper V"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">W/V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for general <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V subset-of-or-equal-to upper W"> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mi>W</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">V \subseteq W</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, are lexicographically shellable. The Möbius function on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W slash upper V"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">W/V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under Bruhat order takes values in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet negative 1 comma 0 comma plus 1 EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ - 1,\,0,\, + 1\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For finite groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, generalized quotients are the same thing as lower intervals in the weak order. This is, however, in general not true. Connections with the weak order are explored and it is shown that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W slash upper V"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">W/V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is always a complete meet-semilattice and a convex order ideal as a subset of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under weak order. Descent classes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript upper I Baseline equals left-brace w element-of upper W vertical-bar l left-parenthesis w s right-parenthesis greater-than l left-parenthesis w right-parenthesis left right double arrow s element-of upper I comma for all s element-of upper S right-brace"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>I</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>w</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>w</mml:mi> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mi>l</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">⇔<!-- ⇔ --></mml:mo> <mml:mi>s</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>I</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>for all</mml:mtext> </mml:mrow> <mml:mspace width="thickmathspace" /> <mml:mi>s</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>S</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{D_I} = \{ w \in W|l(ws) &gt; l(w) \Leftrightarrow s \in I,\;{\text {for all}}\;s \in S\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I subset-of-or-equal-to upper S"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">I \subseteq S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, are also analyzed using generalized quotients. It is shown that each descent class, as a poset under Bruhat order or weak order, is isomorphic to a generalized quotient under the corresponding ordering. The latter half of the paper is devoted to the symmetric group and to t