Lehmer Complexes in Bruhat Intervals

• Lehmer complexes are defined by transporting lower Bruhat intervals to multicomplexes through Lehmer codes, preserving rank data with explicit Poincaré polynomial identities.
• They utilize the Björner–Frankl–Stanley M-complex construction to produce pure, vertex-decomposable, and shellable simplicial complexes that reflect combinatorial properties of Coxeter groups.
• In type A and F4, Lehmer complexes illustrate links between unimodal permutations, smooth Schubert varieties, and the role of weak Lehmer codes in overcoming combinatorial obstructions.

Searching arXiv for the cited Lehmer-complex papers and closely related work. Lehmer complexes are simplicial complexes attached to lower Bruhat intervals by transporting the interval through a Lehmer code to a multicomplex and then applying the Björner–Frankl–Stanley $M$-complex construction. For a finite Coxeter system $(W,S)$ admitting a Lehmer code, each lower interval $[e,w]$ determines an order ideal $J_w$ in a product of chains, and the associated simplicial complex $\Delta_w$ has $f$-polynomial equal to the Poincaré polynomial of the interval. In the framework developed by Bolognini–Sentinelli, these complexes are vertex-decomposable; in type $A$ they encode the Poincaré polynomials of smooth Schubert varieties via unimodal permutations, while in type $F_4$ ordinary Lehmer codes fail to exist but weak Lehmer codes still recover Lehmer complexes interval by interval (Bolognini et al., 6 Jan 2025, Sentinelli et al., 25 Sep 2025).

1. Lehmer codes and the Bruhat-order input

Let $(W,S)$ be a finite Coxeter system with exponents $e_1,\dots,e_n$. A Lehmer code for $(W,S)$0 is a bijection

$(W,S)$1

where $(W,S)$2, such that the inverse

$(W,S)$3

is a poset morphism. Equivalently,

$(W,S)$4

in Bruhat order. In particular,

$(W,S)$5

This condition is the key combinatorial mechanism behind Lehmer complexes: it turns Bruhat intervals into order ideals in a product order (Bolognini et al., 6 Jan 2025).

In type $(W,S)$6, that definition recovers the classical inversion-based Lehmer code

$(W,S)$7

a bijection onto $(W,S)$8 whose inverse preserves Bruhat order. Analogous explicit constructions were given for types $(W,S)$9, $[e,w]$0, and $[e,w]$1 by factoring each element into saturated Bruhat chains and reading off their lengths (Bolognini et al., 6 Jan 2025).

The interval of interest is always a lower Bruhat interval

$[e,w]$2

Once a Lehmer code exists, the interval becomes a combinatorial object in $[e,w]$3, and that order-theoretic translation is what the Lehmer complex formalizes.

2. From lower Bruhat intervals to multicomplexes

Fix a Lehmer code $[e,w]$4. For each $[e,w]$5, the lower interval is sent to

$[e,w]$6

Because the inverse of $[e,w]$7 is order-preserving, $[e,w]$8 is an order ideal. Order ideals in $[e,w]$9 are exactly multicomplexes, so the Bruhat interval is converted into a multicomplex without losing the rank data (Bolognini et al., 6 Jan 2025).

Bolognini–Sentinelli then apply the Björner–Frankl–Stanley $J_w$0-complex construction. For $J_w$1, the complex $J_w$2 is a pure, vertex-colored, shellable simplicial complex of dimension $J_w$3. If $J_w$4 is an order ideal, then $J_w$5 is its $J_w$6-complex. In the Coxeter-theoretic setting,

$J_w$7

Each face $J_w$8 of $J_w$9 inherits a multidegree $\Delta_w$0 by counting how many vertices of each color occur in $\Delta_w$1. The resulting $\Delta_w$2-polynomial is

$\Delta_w$3

Thus the Lehmer complex is not merely attached to the interval: its face enumeration reproduces the interval’s Poincaré, or rank-generating, polynomial exactly (Bolognini et al., 6 Jan 2025).

3. Polynomial identities and topological properties

A classical change of variables relates the $\Delta_w$4- and $\Delta_w$5-polynomials of a $\Delta_w$6-dimensional simplicial complex: $\Delta_w$7 Within the Lehmer-complex construction, one obtains the following statement: for every lower Bruhat interval $\Delta_w$8 in a finite Coxeter group admitting a Lehmer code,

$\Delta_w$9

and moreover $f$0 is vertex-decomposable, hence shellable (Bolognini et al., 6 Jan 2025).

The vertex-decomposability argument proceeds recursively. One peels off a highest-colored vertex whose deletion and link both remain $f$1-complexes of smaller multicomplexes, and continues until the complex reduces to the void or a point. This gives an explicit inductive structure rather than a purely existential shelling argument (Bolognini et al., 6 Jan 2025).

The principal significance of this construction is that Poincaré polynomials of lower Bruhat intervals are realized as $f$2-polynomials of vertex-decomposable simplicial complexes. In the terminology of the paper, the Lehmer complex is therefore a bridge between Bruhat-order combinatorics and a class of simplicial complexes with strong recursive and shellability properties.

4. Type $f$3, unimodal permutations, and smooth Schubert varieties

In type $f$4, Lehmer complexes interact directly with Schubert geometry. For $f$5, the Schubert variety $f$6 is smooth if and only if $f$7 avoids the patterns $f$8 and $f$9. Its Poincaré polynomial factors as

$A$0

where $A$1 is the exponent partition of $A$2 (Bolognini et al., 6 Jan 2025).

A permutation $A$3 is unimodal if there exists $A$4 such that

$A$5

Equivalently, $A$6 avoids $A$7 and $A$8. Bolognini–Sentinelli prove that the set of Poincaré polynomials of all smooth Schubert varieties in $A$9 coincides with

$F_4$0

In particular there are $F_4$1 distinct such polynomials, in bijection with partitions $F_4$2 of size $F_4$3 having strictly decreasing parts (Bolognini et al., 6 Jan 2025).

The small-rank examples make the correspondence concrete. In type $F_4$4, the Lehmer complex of the full group $F_4$5 is $F_4$6, whose $F_4$7-polynomial is

$F_4$8

Among the unimodal permutations in $F_4$9, the element $(W,S)$0 has Lehmer code $(W,S)$1, exponent partition $(W,S)$2, and

$(W,S)$3

These examples illustrate that the Lehmer complex packages the same rank data that appears in Schubert-theoretic factorization formulas (Bolognini et al., 6 Jan 2025).

5. Type $(W,S)$4: failure of ordinary Lehmer codes and the weak replacement

Type $(W,S)$5 provides the main obstruction to a uniform ordinary theory. Let $(W,S)$6 be the Coxeter system of type $(W,S)$7, with exponents $(W,S)$8. If $(W,S)$9 is the longest element, then

$e_1,\dots,e_n$0

Despite this product factorization, the Bruhat order of $e_1,\dots,e_n$1 does not admit a product of chains as subposet. Equivalently, there is no bijection

$e_1,\dots,e_n$2

whose inverse preserves Bruhat order. This answers negatively, in type $e_1,\dots,e_n$3, the question asked by Billey, Fan and Losonczy (Sentinelli et al., 25 Sep 2025).

The proof strategy is combinatorial and obstruction-based. One checks that any order-preserving immersion would have to send the rank-$e_1,\dots,e_n$4 elements bijectively onto the $e_1,\dots,e_n$5 tuples in

$e_1,\dots,e_n$6

Up to the automorphism group of $e_1,\dots,e_n$7, generated by inversion $e_1,\dots,e_n$8 and the Dynkin-diagram symmetry $e_1,\dots,e_n$9, there are $(W,S)$00 such immersions. For each candidate, some rank-$(W,S)$01 tuple has a set of lower covers whose images do not admit a common upper cover in Bruhat order, so no consistent extension to level $(W,S)$02 exists (Sentinelli et al., 25 Sep 2025).

To recover the interval-by-interval multicomplex construction, the paper introduces a weak Lehmer code. For a finite-rank Coxeter system with $(W,S)$03, a finite collection

$(W,S)$04

of rank-preserving injections $(W,S)$05 is a weak Lehmer code if two conditions hold. First, whenever the set of maximal elements

$(W,S)$06

is a singleton, then $(W,S)$07 is an order ideal in $(W,S)$08. Second, for every $(W,S)$09, some $(W,S)$10, up to automorphism of $(W,S)$11, makes $(W,S)$12 an order ideal. A classical Lehmer code is the special case $(W,S)$13, with bijectivity onto the full product of chains (Sentinelli et al., 25 Sep 2025).

In type $(W,S)$14, exactly two maps $(W,S)$15 suffice. The construction uses a parabolic decomposition into saturated Bruhat chains

$(W,S)$16

$(W,S)$17

$(W,S)$18

together with chains $(W,S)$19, each of length $(W,S)$20 or $(W,S)$21, satisfying

$(W,S)$22

For a factorization $(W,S)$23 with $(W,S)$24 and $(W,S)$25,

$(W,S)$26

and

$(W,S)$27

These maps are injective and rank-preserving, and for every lower interval one of them, up to automorphism, yields the required order ideal. The pair $(W,S)$28 is therefore a weak Lehmer code for $(W,S)$29. Once

$(W,S)$30

is an order ideal with $(W,S)$31-polynomial $(W,S)$32, the general machinery produces a balanced vertex-decomposable simplicial complex, called the Lehmer complex of $(W,S)$33, whose $(W,S)$34-polynomial is again $(W,S)$35 (Sentinelli et al., 25 Sep 2025).

6. Examples, further structure, and terminological scope

The $(W,S)$36 examples show the construction in explicit form. For

$(W,S)$37

one has $(W,S)$38, $(W,S)$39 is fixed by every Dynkin automorphism, and

$(W,S)$40

The lower interval $(W,S)$41 has $(W,S)$42 elements. Its corresponding multicomplex $(W,S)$43 has maximal elements

$(W,S)$44

and the associated Lehmer complex is a pure simplicial complex on seven vertices, one for each coatom, with $(W,S)$45 facets. For the shorter element

$(W,S)$46

one finds

$(W,S)$47

Here $(W,S)$48 is an order ideal generated by the single maximal chain $(W,S)$49, so $(W,S)$50 is isomorphic to $(W,S)$51, and the Lehmer complex is a direct product of four chains of lengths $(W,S)$52 (Sentinelli et al., 25 Sep 2025).

The $(W,S)$53 theory also supplies additional order-theoretic structure. Every lower interval $(W,S)$54 has rank sequence an $(W,S)$55-sequence, realized by the weak code as the $(W,S)$56-vector of an explicit multicomplex in $(W,S)$57. The Lehmer complex of each interval is balanced and vertex-decomposable, hence Cohen–Macaulay, and its $(W,S)$58-polynomial is exactly the Bruhat–Poincaré polynomial $(W,S)$59. If $(W,S)$60 is $(W,S)$61-principal, meaning that $(W,S)$62 is a singleton, then the set $(W,S)$63 of $(W,S)$64-principal elements is a meet-semilattice under Bruhat order, identified with the componentwise order on $(W,S)$65. Likewise, the $(W,S)$66-unimodal elements form a lattice $(W,S)$67 isomorphic to a sublattice of $(W,S)$68; in type $(W,S)$69, $(W,S)$70 is precisely in bijection with the set $(W,S)$71 of palindromic Poincaré polynomials of Schubert varieties, ordered by reverse inclusion of inversion sets (Sentinelli et al., 25 Sep 2025).

A common terminological confusion arises from the word “complex.” In the combinatorial setting considered here, a Lehmer complex is a simplicial complex attached to a Bruhat interval. By contrast, the neural-network paper “Efficient and Interpretable Neural Networks Using Complex Lehmer Transform” studies a weighted Lehmer transform with complex-valued parameter $(W,S)$72 and introduces complex Lehmer activation units; in that context, “complex” refers to the complex domain and to phase-sensitive behavior, not to a simplicial complex (Ataei et al., 25 Jan 2025). This distinction is purely terminological, but it separates two mathematically unrelated uses of the word “Lehmer” in current arXiv literature.