Weak Lehmer Codes in F4 and Beyond

• Weak Lehmer codes are finite families of injective, rank-preserving encodings that locally capture inversion counts and generate multicomplex models for Bruhat intervals.
• They generalize the classical Lehmer code by relaxing global order preservation, as seen in type F4 where no single product-of-chains embedding exists.
• The construction uses parabolic decompositions and saturated chains to yield explicit Lehmer complexes, offering insights into palindromicity and interval invariants.

Searching arXiv for the cited papers to ground the article in current records. Weak Lehmer codes are finite families of injective, rank-preserving encodings that replace the single global order-preserving product-of-chains parametrization required by a classical Lehmer code. In the classical setting of permutations, the Lehmer code records inversion counts and identifies $S_n$ with a rectangular product of chains. In more general Coxeter-theoretic settings, especially for lower Bruhat intervals, analogous factorizations of rank-generating functions often suggest such a structure, but that suggestion can fail at the level of order embedding. The paper "A weak Lehmer code for type $F_4$" establishes precisely this phenomenon for $F_4$: although the full Poincaré polynomial factors as a product of $q$-analogues, no global Lehmer code exists, yet a weaker local theory still yields multicomplexes and Lehmer complexes for every lower Bruhat interval (Sentinelli et al., 25 Sep 2025).

1. Classical origin and the meaning of “weak”

For a permutation $\sigma \in S_n$, the classical Lehmer code is

$$c(\sigma)=(c_1(\sigma),\dots,c_n(\sigma)), \qquad c_i(\sigma)=|\{j>i:\sigma_j<\sigma_i\}|.$$

It encodes the inversion set of $\sigma$ and gives a bijection

$S_n \to \prod_{i=1}^n [n-i]_0,$

where $[m]_0=\{0,1,\dots,m\}$. Under componentwise order on the target, the inverse map is order-preserving, so the image is a product-of-chains subposet whose rank-generating function matches that of $S_n$ (Sentinelli et al., 25 Sep 2025).

This classical picture separates into two logically distinct features. One is numerical: the rank-generating function factors as a product of $F_4$0-analogues. The other is order-theoretic: there is an order-preserving inverse from a rectangular box of code vectors into the ambient poset. The weak Lehmer code formalism retains the first kind of control locally on intervals while relaxing the second globally. In type $F_4$1, this relaxation is necessary because the Bruhat order does not admit a product of chains as a subposet, even though the Poincaré polynomial factors (Sentinelli et al., 25 Sep 2025).

A useful contrast appears in type $F_4$2. In the symmetric group, several Lehmer-code-derived constructions model weak-order phenomena without giving a global code for the full weak order. Denoncourt showed that the Lehmer codes of permutations in a weak-order interval form a distributive lattice under the product order, and that its rank-generating function matches that of the interval (Denoncourt, 2011). In a different direction, the consecutive Lehmer code for parabolic quotients of $F_4$3 is not injective on the full parabolic quotient, but on $F_4$4-avoiding elements it becomes bijective and componentwise order realizes the parabolic Tamari lattice (Fang et al., 2020). This suggests that “weak” may mean either intervalwise distributive refinement or quotientwise order realization, depending on context. In type $F_4$5, the weak Lehmer code of (Sentinelli et al., 25 Sep 2025) is a further generalization: a finite family of codes suffices to realize every lower Bruhat interval as a multicomplex, possibly after precomposition by an automorphism.

2. The $F_4$6 obstruction to a strong Lehmer code

Let $F_4$7 be the finite Weyl/Coxeter group of rank $F_4$8 with simple reflections $F_4$9 and Coxeter graph

$F_4$0

where “$F_4$1” denotes $F_4$2. Bruhat order $F_4$3 on $F_4$4 is graded by the length function $F_4$5, and for $F_4$6 the lower interval $F_4$7 is $F_4$8 (Sentinelli et al., 25 Sep 2025).

The exponents of $F_4$9 are $q$0. Writing $q$1, the full group’s rank-generating function factors as

$q$2

Such a factorization guarantees a rank-preserving bijection to a product of chains, but it does not by itself provide an order-preserving inverse. The negative result of (Sentinelli et al., 25 Sep 2025) shows that this distinction is essential in type $q$3.

The main theorem states that the Bruhat order of $q$4 does not admit a product of chains as a subposet; equivalently, $q$5 does not admit a Lehmer code in the sense of an order-preserving bijection

$q$6

with rank compatibility (Sentinelli et al., 25 Sep 2025). This answers negatively, in type $q$7, the question of Billey, Fan and Losonczy whether a rank-symmetric lower Bruhat interval admitting a product factorization of its rank-generating function also admits such a Lehmer code.

The obstruction is formulated through truncated rank layers. Let

$q$8

For $q$9, set

$\sigma \in S_n$0

Then

$\sigma \in S_n$1

The authors compute all injective poset morphisms $\sigma \in S_n$2 and reduce them up to the automorphism group

$\sigma \in S_n$3

generated by inversion $\sigma \in S_n$4 and the diagram automorphism $\sigma \in S_n$5 (Sentinelli et al., 25 Sep 2025). They then attempt to extend these immersions to rank $\sigma \in S_n$6. For each coatom-set configuration arising from an element of rank $\sigma \in S_n$7 in $\sigma \in S_n$8, one must find a corresponding upper bound in $\sigma \in S_n$9; for every candidate immersion, some required configuration has no such upper bound. Therefore no order-preserving bijection onto $$c(\sigma)=(c_1(\sigma),\dots,c_n(\sigma)), \qquad c_i(\sigma)=|\{j>i:\sigma_j<\sigma_i\}|.$$0 exists.

A plausible implication is that factorizations of Poincaré polynomials in exceptional types should not be interpreted as evidence for global rectangular order models without additional structural input. In type $$c(\sigma)=(c_1(\sigma),\dots,c_n(\sigma)), \qquad c_i(\sigma)=|\{j>i:\sigma_j<\sigma_i\}|.$$1, the factorization survives, but the order-embedding fails.

3. Formal definition of weak Lehmer codes

Let $$c(\sigma)=(c_1(\sigma),\dots,c_n(\sigma)), \qquad c_i(\sigma)=|\{j>i:\sigma_j<\sigma_i\}|.$$2 be a Coxeter system of finite rank $$c(\sigma)=(c_1(\sigma),\dots,c_n(\sigma)), \qquad c_i(\sigma)=|\{j>i:\sigma_j<\sigma_i\}|.$$3. A finite set

$$c(\sigma)=(c_1(\sigma),\dots,c_n(\sigma)), \qquad c_i(\sigma)=|\{j>i:\sigma_j<\sigma_i\}|.$$4

is a weak Lehmer code if the following conditions hold (Sentinelli et al., 25 Sep 2025):

1. Each $$c(\sigma)=(c_1(\sigma),\dots,c_n(\sigma)), \qquad c_i(\sigma)=|\{j>i:\sigma_j<\sigma_i\}|.$$5 is injective and rank-preserving, meaning that $$c(\sigma)=(c_1(\sigma),\dots,c_n(\sigma)), \qquad c_i(\sigma)=|\{j>i:\sigma_j<\sigma_i\}|.$$6 is the sum of the coordinates of $$c(\sigma)=(c_1(\sigma),\dots,c_n(\sigma)), \qquad c_i(\sigma)=|\{j>i:\sigma_j<\sigma_i\}|.$$7.
2. For any $$c(\sigma)=(c_1(\sigma),\dots,c_n(\sigma)), \qquad c_i(\sigma)=|\{j>i:\sigma_j<\sigma_i\}|.$$8 and any $$c(\sigma)=(c_1(\sigma),\dots,c_n(\sigma)), \qquad c_i(\sigma)=|\{j>i:\sigma_j<\sigma_i\}|.$$9, if

$\sigma$0

in the componentwise order, then

$\sigma$1

is a multicomplex, that is, an order ideal of $\sigma$2.

1. For every $\sigma$3, there exist $\sigma$4 and an automorphism $\sigma$5 such that

$\sigma$6

is a multicomplex.

This definition relaxes the single global product-of-chains embedding in two ways. First, it allows a finite family rather than one code. Second, it requires the multicomplex property only intervalwise, possibly after precomposition by an automorphism (Sentinelli et al., 25 Sep 2025). What is retained is sufficient for the rank-generating function of every lower Bruhat interval to be realized as the Hilbert series of a multicomplex and therefore to feed the canonical Lehmer-complex construction of [BS, Section 2], as quoted in (Sentinelli et al., 25 Sep 2025).

The $\sigma$7 theory also introduces two derived classes of elements relative to a fixed $\sigma$8. An element $\sigma$9 is $S_n \to \prod_{i=1}^n [n-i]_0,$0-principal if

$S_n \to \prod_{i=1}^n [n-i]_0,$1

In that case,

$S_n \to \prod_{i=1}^n [n-i]_0,$2

hence

$S_n \to \prod_{i=1}^n [n-i]_0,$3

The set $S_n \to \prod_{i=1}^n [n-i]_0,$4 of $S_n \to \prod_{i=1}^n [n-i]_0,$5-principal elements forms a meet-semilattice isomorphic to its coordinate image under $S_n \to \prod_{i=1}^n [n-i]_0,$6 (Sentinelli et al., 25 Sep 2025).

For $S_n \to \prod_{i=1}^n [n-i]_0,$7, let

$S_n \to \prod_{i=1}^n [n-i]_0,$8

Then $S_n \to \prod_{i=1}^n [n-i]_0,$9 is $[m]_0=\{0,1,\dots,m\}$0-unimodal if $[m]_0=\{0,1,\dots,m\}$1 is the lexicographically minimal element in $[m]_0=\{0,1,\dots,m\}$2. The induced Bruhat order on $[m]_0=\{0,1,\dots,m\}$3 is isomorphic to componentwise order on $[m]_0=\{0,1,\dots,m\}$4 (Sentinelli et al., 25 Sep 2025). These notions isolate strata on which the weak code behaves particularly rigidly.

4. Explicit construction in type $[m]_0=\{0,1,\dots,m\}$5

The construction in (Sentinelli et al., 25 Sep 2025) is based on a parabolic decomposition relative to

$[m]_0=\{0,1,\dots,m\}$6

Five saturated Bruhat chains are fixed:

• $[m]_0=\{0,1,\dots,m\}$7.
• $[m]_0=\{0,1,\dots,m\}$8.
• $[m]_0=\{0,1,\dots,m\}$9.
• $S_n$0, a saturated chain of length $S_n$1 beginning at $S_n$2 and ending at $S_n$3.
• $S_n$4, a saturated chain beginning at $S_n$5 and ending at $S_n$6.

These chains realize the parabolic structure:

$S_n$7

the parabolic subgroup of type $S_n$8 generated by $S_n$9, while the minimal left coset representatives for $F_4$00 decompose into $F_4$01. Consequently,

$F_4$02

and every element has a unique factorization

$F_4$03

with $F_4$04, $F_4$05, and

$F_4$06

(Sentinelli et al., 25 Sep 2025).

Using this factorization, two maps $F_4$07 are defined. For $F_4$08,

$F_4$09

and

$F_4$10

Both maps are injective and rank-preserving because each chain is linear and has a unique element of each length (Sentinelli et al., 25 Sep 2025).

Their images are described explicitly as unions of boxes in $F_4$11:

Map	Image description
$F_4$12	$F_4$13
$F_4$14	$F_4$15

These sit inside the ambient product $F_4$16 under componentwise order (Sentinelli et al., 25 Sep 2025).

Theoremally, the set $F_4$17 is a weak Lehmer code for type $F_4$18: condition (2) holds whenever the image on an interval has a unique maximal element, and condition (3) holds globally after allowing automorphisms generated by inversion and the diagram automorphism (Sentinelli et al., 25 Sep 2025). This gives an explicit encoding and decoding mechanism for every element of $F_4$19, without asserting the existence of a single rectangular order embedding.

5. Multicomplexes, Lehmer complexes, and interval invariants

A multicomplex is an order ideal $F_4$20 under componentwise order. Its $F_4$21-polynomial is

$F_4$22

For type $F_4$23, given any $F_4$24, one chooses $F_4$25 and $F_4$26 as guaranteed by the weak-code definition and sets

$F_4$27

Then $F_4$28 is a multicomplex and

$F_4$29

where $F_4$30 is the rank-generating function of the interval $F_4$31 (Sentinelli et al., 25 Sep 2025).

This is the principal structural payoff of weak Lehmer codes. The global order-embedding is unavailable, but every lower Bruhat interval still acquires an explicit multicomplex model. By the canonical construction described in [BS, Section 2], and quoted in (Sentinelli et al., 25 Sep 2025), such a multicomplex produces a balanced vertex-decomposable simplicial complex whose $F_4$32-polynomial is the interval rank-generating function. In the terminology of (Sentinelli et al., 25 Sep 2025), this yields the Lehmer complex of $F_4$33 in type $F_4$34.

The behavior of principal and unimodal elements further refines this picture. If $F_4$35 is $F_4$36-principal, then the interval image is the full box below $F_4$37, and the rank-generating function factors as a product of $F_4$38-analogues:

$F_4$39

For $F_4$40, the set of palindromic Poincaré polynomials is realized by $F_4$41-unimodal elements together with the longest element:

$F_4$42

Moreover, both $F_4$43 and $F_4$44 are lattices under Bruhat order, isomorphic to their coordinate images under $F_4$45. By contrast, $F_4$46 is strictly smaller than $F_4$47 and does not form a lattice (Sentinelli et al., 25 Sep 2025).

This suggests that weak Lehmer codes do more than recover Hilbert series: they stratify the Bruhat order into regions where factorization, palindromicity, and lattice structure are simultaneously visible in coordinates.

6. Examples and comparison with earlier Lehmer-code frameworks

Several explicit examples in (Sentinelli et al., 25 Sep 2025) illustrate how weak Lehmer codes behave beyond the principal-box case. Let

$F_4$48

This element is fixed by $F_4$49, and

$F_4$50

The multicomplex

$F_4$51

has maximal elements

$F_4$52

The associated Lehmer complex of $F_4$53 has $F_4$54 facets, and its generating function agrees with $F_4$55 (Sentinelli et al., 25 Sep 2025). This is a genuinely nontrivial interval: the code image is a multicomplex with several maximal elements rather than a single principal box.

A second example concerns principal and unimodal behavior. Let $F_4$56 and $F_4$57. Then

$F_4$58

so

$F_4$59

and $F_4$60. Its orbit by palindromic factorization is

$F_4$61

with images

$F_4$62

The lexicographically minimal one is $F_4$63, so $F_4$64 (Sentinelli et al., 25 Sep 2025).

For the longest element $F_4$65,

$F_4$66

Although this factors as a product of $F_4$67-analogues, there is no global strong Lehmer code embedding onto $F_4$68 with order-preserving inverse. Nevertheless, the weak-code framework still produces a multicomplex recovering $F_4$69 (Sentinelli et al., 25 Sep 2025). This sharply distinguishes factorization of generating functions from existence of a strong code.

Relative to earlier Lehmer-code constructions, the $F_4$70 framework occupies a distinct position. Denoncourt’s interval theory in type $F_4$71 shows that the Lehmer codes of permutations in a weak-order interval form a distributive lattice, with a canonical base poset of join-irreducibles (Denoncourt, 2011). Tomie’s analysis of Denoncourt’s base posets studies when these interval-derived posets are $F_4$72-free, characterizing that condition by avoidance of the patterns $F_4$73 and $F_4$74 (Tomie, 2011). In the parabolic type-$F_4$75 setting, the consecutive Lehmer code realizes the parabolic Tamari lattice on $F_4$76-avoiding elements via componentwise order (Fang et al., 2020). The weak Lehmer code for $F_4$77 differs from all of these in that it is designed for Bruhat order in an exceptional Weyl group, it requires a finite family of maps rather than one, and its target is not a single product-of-chains image but a collection of intervalwise multicomplexes (Sentinelli et al., 25 Sep 2025).

7. Broader significance, limitations, and open problems

The strong/weak distinction is explicit in (Sentinelli et al., 25 Sep 2025). A strong Lehmer code requires a single bijection

$F_4$78

whose inverse is order-preserving, so the image is a product-of-chains subposet. A weak Lehmer code relaxes this to a finite family $F_4$79, each injective and rank-preserving, such that for every $F_4$80 there is at least one $F_4$81, possibly composed with an automorphism, for which the image of $F_4$82 is a multicomplex (Sentinelli et al., 25 Sep 2025).

What is retained under this relaxation is substantial. Weak codes still realize rank-generating functions as multicomplex Hilbert series, provide explicit combinatorial encoding and decoding via parabolic chains, and yield poset isomorphisms on principal and unimodal strata (Sentinelli et al., 25 Sep 2025). What is lost is equally clear: there is no single global order-embedding onto a rectangular box, and there is no uniform surjectivity onto one product $F_4$83.

The paper places the $F_4$84 result in a wider landscape. In classical types $F_4$85, $F_4$86, and $F_4$87, strong Lehmer codes exist, and thus multicomplexes for all lower intervals are already available (Sentinelli et al., 25 Sep 2025). By contrast, a Coxeter system of type $F_4$88 does not admit any weak Lehmer code, and universal Coxeter systems with $F_4$89 do not admit weak codes because some lower intervals have rank sequences that are not $F_4$90-sequences, for example $F_4$91 (Sentinelli et al., 25 Sep 2025). The $F_4$92 construction therefore sits between positive classical results and stronger negative results where even weak local control fails.

The exceptional types $F_4$93, $F_4$94, and $F_4$95 remain open. The construction for $F_4$96 suggests a possible route through carefully chosen parabolic factorizations and canonical chains partitioning $F_4$97 into a small number of blocks, but obstacles remain in ensuring conditions (2) and (3) uniformly, especially uniqueness of maxima and the multicomplex ideal property (Sentinelli et al., 25 Sep 2025). The same source also identifies further problems: classifying $F_4$98-principal and $F_4$99-unimodal elements in exceptional types, optimizing the number $F_4$00 of maps in a weak code family, and relating weak-code images to pattern-avoidance and rational smoothness through palindromicity (Sentinelli et al., 25 Sep 2025).

This suggests a general interpretation of weak Lehmer codes as a compromise between algebraic factorization and order-theoretic rigidity. They do not recover a global rectangular model of Bruhat order, but they preserve enough structure to encode every lower interval by multicomplex data, thereby extending the Lehmer-code philosophy into settings where the classical product-of-chains paradigm provably breaks down.