Poset of Parabolic Cosets

• The poset of parabolic cosets is an order-theoretic structure defined by the inclusion of cosets in finite Coxeter groups using their intersection lattices.
• It connects Coxeter complex theory, Bruhat order, and Schubert geometry, offering topological insights and representation-theoretic invariants like homology characters.
• Extensions to double coset systems and symmetric groups provide combinatorial, enumerative, and probabilistic tools for analyzing parabolic subgroup structures.

The poset of parabolic cosets is an order-theoretic structure attached to parabolic subgroups in Coxeter-type and Artin-type settings. In the finite Coxeter case, if $W$ is a finite Coxeter group with intersection lattice $L$ of reflecting hyperplanes, the basic object is

$$P=\biguplus_{X\in L} W/W_X=\{\,wW_X : w\in W,\ X\in L\,\},$$

ordered by inclusion of subsets, where

$W_X=\{\,w\in W : X\subseteq \mathrm{Fix}(w)\,\}.$

This poset sits at the intersection of Coxeter complexes, Bruhat order, Schubert geometry, and representation theory. Closely related, but distinct, order structures arise from parabolic double cosets, from intersections of Bruhat intervals with parabolic cosets, and from inclusion posets of parabolic subgroups and cosets in Artin groups (Douvropoulos et al., 15 Sep 2025, Billey et al., 2016, Oh et al., 2022).

1. Definitions and competing order conventions

For a finite Coxeter group $W$ of rank $n$, the parabolic coset poset $P$ is defined by inclusion of cosets $wW_X$, with $X$ ranging over the intersection lattice $L$ of the reflecting arrangement. The group $L$0 acts on $L$1 by left multiplication. The interval $L$2 in $L$3 is isomorphic to $L$4, the order ideal below any element $L$5 is itself the parabolic coset poset for $L$6, $L$7 has a unique maximum $L$8, and its minimal elements are the singletons $L$9 (Douvropoulos et al., 15 Sep 2025).

A classical precursor is the Coxeter complex: for a Coxeter system $$P=\biguplus_{X\in L} W/W_X=\{\,wW_X : w\in W,\ X\in L\,\},$$0, the set of ordinary one-sided cosets $$P=\biguplus_{X\in L} W/W_X=\{\,wW_X : w\in W,\ X\in L\,\},$$1, with $$P=\biguplus_{X\in L} W/W_X=\{\,wW_X : w\in W,\ X\in L\,\},$$2, forms the Coxeter complex of $$P=\biguplus_{X\in L} W/W_X=\{\,wW_X : w\in W,\ X\in L\,\},$$3. This one-sided theory is substantially older and better understood than the double-coset theory (Billey et al., 2016).

A recurring source of ambiguity is that the literature studies several related posets under nearby names.

Setting	Elements	Order
Finite Coxeter parabolic coset poset	$$P=\biguplus_{X\in L} W/W_X=\{\,wW_X : w\in W,\ X\in L\,\},$$4	Inclusion
Finite Coxeter parabolic double-coset system	$$P=\biguplus_{X\in L} W/W_X=\{\,wW_X : w\in W,\ X\in L\,\},$$5	Reverse containment
Symmetric-group contingency-table model	Tables for $$P=\biguplus_{X\in L} W/W_X=\{\,wW_X : w\in W,\ X\in L\,\},$$6	Majorization or related containment/intersection orders

The double-coset analogue is not merely a rephrasing of the one-sided poset. For finite Coxeter groups, the set $$P=\biguplus_{X\in L} W/W_X=\{\,wW_X : w\in W,\ X\in L\,\},$$7 of all parabolic double cosets $$P=\biguplus_{X\in L} W/W_X=\{\,wW_X : w\in W,\ X\in L\,\},$$8 is equipped with the partial order

$$P=\biguplus_{X\in L} W/W_X=\{\,wW_X : w\in W,\ X\in L\,\},$$9

that is, reverse containment (Kobayashi, 2019). In the symmetric group, parabolic double cosets are also modeled by contingency tables, and the resulting collection carries a majorization order with explicit probabilistic meaning (Diaconis et al., 2021).

2. Topology and representation theory of the finite Coxeter poset

The most developed global theory concerns the finite Coxeter poset $W_X=\{\,w\in W : X\subseteq \mathrm{Fix}(w)\,\}.$0. It is Cohen–Macaulay, and for the proper part $W_X=\{\,w\in W : X\subseteq \mathrm{Fix}(w)\,\}.$1, the unique nonzero reduced homology group occurs in degree $W_X=\{\,w\in W : X\subseteq \mathrm{Fix}(w)\,\}.$2. If $W_X=\{\,w\in W : X\subseteq \mathrm{Fix}(w)\,\}.$3 carries character $W_X=\{\,w\in W : X\subseteq \mathrm{Fix}(w)\,\}.$4, then

$W_X=\{\,w\in W : X\subseteq \mathrm{Fix}(w)\,\}.$5

where $W_X=\{\,w\in W : X\subseteq \mathrm{Fix}(w)\,\}.$6 is the Möbius function of the intersection lattice and $W_X=\{\,w\in W : X\subseteq \mathrm{Fix}(w)\,\}.$7 is the permutation character of $W_X=\{\,w\in W : X\subseteq \mathrm{Fix}(w)\,\}.$8 acting on $W_X=\{\,w\in W : X\subseteq \mathrm{Fix}(w)\,\}.$9 (Douvropoulos et al., 15 Sep 2025).

This formula has a conjugacy-class version. If $W$0 is a set of representatives for the $W$1-orbits in $W$2, then

$W$3

The same work shows that after tensoring with the sign character $W$4, the homology character becomes positive in the natural basis of the parabolic Burnside ring: $W$5 That positivity is tied to the positive chamber complex $W$6, defined by taking Weyl chambers lying on the positive side of a generic hyperplane. In this interpretation, the coefficients of $W$7 encode the colored $W$8-vector of $W$9, while the untwisted expression reflects $n$0-vector data (Douvropoulos et al., 15 Sep 2025).

The representation-theoretic description also admits a chamber-combinatorial form. One has

$n$1

with notation as in the source. This places the poset of parabolic cosets in direct contact with ascent sets, shellings, and chamber enumeration (Douvropoulos et al., 15 Sep 2025).

3. Parabolic cosets inside Bruhat order

A second line of work studies not the global poset $n$2, but the induced order structure obtained by slicing Bruhat intervals by parabolic cosets. For a Coxeter group $n$3, a subset $n$4, and $n$5, one has the decomposition

$n$6

If $n$7 and $n$8, then $n$9 has a unique maximal element in Bruhat order (Oh et al., 2022).

This uniqueness extends much further. In an arbitrary Coxeter group, if $P$0 is nonempty, then it is itself a Bruhat interval with a unique minimal element and a unique maximal element. More precisely, there exist unique $P$1 such that

$P$2

Thus the induced poset on a parabolic coset inside a Bruhat interval is never fragmented: it is always interval-like (Marietti, 2022).

The unique-maximal-element property has concrete enumerative and geometric consequences. For the Poincaré polynomial

$P$3

one obtains

$P$4

where $P$5 is defined from the unique maximal element of $P$6. In geometric terms, the fibers of standard parabolic projection maps on Schubert varieties are themselves Schubert varieties, and in the BP decomposition case all fibers are isomorphic (Oh et al., 2022).

A plausible implication is that the interval structure of parabolic slices provides a local regularity principle complementing the global Cohen–Macaulay theory of the full parabolic coset poset.

4. Parabolic double cosets and reverse-containment posets

For a Coxeter system $P$7, a parabolic double coset is a subset of the form

$P$8

Every such double coset contains a unique minimal-length representative, but the same double coset may admit many different presentations $P$9. To remove that ambiguity, the theory of lex-minimal presentation selects the unique presentation for which $wW_X$0 is minimal in the coset and $wW_X$1 is lexicographically minimal (Billey et al., 2016).

The reverse-containment poset $wW_X$2 of parabolic double cosets provides a two-sided analogue of the Coxeter complex. It decomposes as

$wW_X$3

where $wW_X$4 consists of those double cosets whose maximal element is $wW_X$5. Each $wW_X$6 is a connected component in the Hasse diagram, maximal elements of $wW_X$7 correspond bijectively to elements of $wW_X$8, and each double coset $wW_X$9 contains unique minimal and maximal elements $X$0 and $X$1 in Bruhat order. Moreover,

$X$2

so each double coset is a two-sided weak-order interval (Kobayashi, 2019).

The order structure is refined by a local dimension function on each component: $X$3 where $X$4 is the number of weak coatoms of $X$5. The minimal element of $X$6 is $X$7, and the singleton $X$8 is the maximal element. In addition, every parabolic double coset is regular in the Bruhat graph (Kobayashi, 2019).

Enumeration is closely tied to this structural theory. Lex-minimal presentations are used in a finite automaton approach to count parabolic double cosets, and for $X$9 the counting problem is expressed through the marine model and the $L$0-ocean (Billey et al., 2016). This suggests that order-theoretic rigidity and algorithmic enumerability are unusually tightly linked in the double-coset setting.

5. The symmetric group: contingency tables, weak orders, and majorization

In the symmetric group, parabolic subgroups are Young subgroups

$L$1

for a partition $L$2. For two partitions $L$3 and $L$4, the double cosets

$L$5

are in bijection with $L$6 contingency tables $L$7 of nonnegative integers with prescribed row sums $L$8 and column sums $L$9. The size of the double coset corresponding to $L$00 is

$L$01

and the induced probability distribution from the uniform measure on $L$02 is the Fisher–Yates distribution

$L$03

Under the majorization order on tables with fixed margins, more balanced tables are more probable (Diaconis et al., 2021).

This contingency-table realization equips the set of parabolic double cosets with a concrete poset model. One formulation emphasizes majorization; another emphasizes containment/intersection of cosets and its relationship with maximal contingency tables and weak orders. In particular, Browning identifies $L$04, the number of distinct parabolic double cosets in $L$05, with the number of maximal two-way contingency tables with sum $L$06, derives an explicit formula for $L$07, proves that the formula yields a polynomial-time algorithm, computes $L$08 for $L$09, and establishes the asymptotic

$L$10

(Browning, 2020).

The weak-order viewpoint is also explicit. The same work relates parabolic double cosets to pairs of weak orders with no consecutive embeddings and writes

$L$11

where $L$12 counts such pairs of weak orders (Browning, 2020). Together, these results make the symmetric group the setting in which the combinatorics of the parabolic double-coset poset is most explicitly modeled.

6. Artin groups, braid groups, and adjacent generalizations

In large-type Artin groups, the order theory of parabolic subgroups has a geometric realization. The Artin complex $L$13 is a simplicial complex whose vertices and simplices correspond to proper parabolic subgroups, and its first barycentric subdivision is the geometric realization of the poset of parabolic subgroups ordered by inclusion. If $L$14 is of large type and $L$15, then $L$16 is systolic. In the same setting, the intersection of any collection of parabolic subgroups is again parabolic, the set of all parabolic subgroups forms a lattice under inclusion, and every subset of $L$17 is contained in a unique minimal parabolic subgroup (Cumplido et al., 2020).

A complementary approach uses retractions to standard parabolic subgroups in Artin groups. For $L$18, one has left and right retractions $L$19 and $L$20, and every $L$21 admits a unique decomposition

$L$22

where $L$23 is $L$24-reduced. The map $L$25 gives a transversal of $L$26-right cosets, the collection of right parabolic cosets $L$27 carries a poset structure by inclusion, and the retractions satisfy

$L$28

Under suitable assumptions, there is also a unique double-coset representative with both left and right retractions trivial (Digne et al., 2024).

For braid groups, the situation is more limited from the poset-theoretic standpoint. The double coset problem is unsolvable for subgroups of $L$29 when $L$30, but for parabolic subgroups with connected associated Coxeter graph it is solvable, and the result was later generalized to all parabolic subgroups of braid groups. The proof reduces the problem to simultaneous conjugacy, uses explicit shift elements $L$31, and relies on the double-centralizer formula

$L$32

for connected parabolic subgroups $L$33. However, this work does not explicitly discuss the poset or lattice structure of the set of parabolic double cosets in $L$34 (Kalka et al., 2014).

Outside the group setting, the contrast becomes sharper. For a finite loop $L$35 with subloop $L$36, the collections

$L$37

are meet-semilattices under inclusion. In loops with the antiautomorphic inverse property, the map $L$38 induces an isomorphism

$L$39

This does not define parabolic cosets in the Coxeter sense, but it isolates a general phenomenon: once coset intersections become nontrivial, the resulting posets can encode substantial geometry and combinatorics (Kinyon et al., 2011).

The modern literature therefore uses the expression “poset of parabolic cosets” in several closely related but non-identical senses. In finite Coxeter groups it denotes a well-structured inclusion poset with strong topological and representation-theoretic invariants; in Bruhat theory it describes interval-like slices by parabolic cosets; in double-coset theory it leads to reverse-containment complexes and contingency-table models; and in Artin groups it interacts with retractions, lattice structures, and systolic geometry.