Imaginary standard Lyndon periodicity in untwisted affine types

Pranava Kumar, Abhijay Gangarapu

Editorial revision, September 2026 · MSC 2020: 17B67, 17B22, 05E10

Figure 1 · The insertion formula, letter by letter

exact words produced by the supplementary classical constructor

initial prefix $L_i$: —
inserted period $W_i$: —
final suffix $R_i$: —
period parent: —

The word grows only by insertion. For each displayed family, $\SL_i(k\delta)=L_iW_i^{k-1}R_i$. Blue letters form the fixed initial prefix, graphite groups are copies of the period, and green letters form the final suffix. The two degree-$2\delta$ lists were also reproduced by a separate exact Lyndon-bracket checker.

Abstract

A fixed insertion law in every untwisted type

We prove that imaginary standard Lyndon words satisfy a fixed insertion formula in every untwisted affine type and every alphabet order, by establishing the remaining classical cases uniformly in the rank and combining them with the exceptional-type verification of Elkins and Tsymbaliuk. A type-independent phase argument identifies the exact increasing-chain preperiod with the initial standard prefix followed by a proper suffix of the period's Lyndon parent. This gives the bound $2|\delta|-2$ for its length and recovers the increasing positive system and its Dynkin parent tree from the two lowest imaginary degrees.

The classical argument represents real blocks as greatest extensions of chain or fork ideals. A first-endpoint comparison and a reflected raising lemma determine the word in degree $2\delta$, after which an existing propagation theorem gives every degree. We also give polynomial preprocessing in every untwisted type, with a sharper classical bound, and determine the primitive column lattices and zero-weight switch bounds in the classical models.

Definitions

Words that remember a basis position

Order the simple roots of an affine Lie algebra. A Lyndon word is a nonempty word smaller than each of its nonempty proper suffixes in the resulting lexicographic order. The standard Lyndon construction assigns selected words to positive root spaces and obtains a basis by recursive Lie brackets.

Real root spaces are one-dimensional, so a real root has one standard word. The imaginary root $k\delta$ has multiplicity equal to the finite rank $n$ in the untwisted case. We therefore list its words in decreasing order,

\[ \SL_1(k\delta)>\cdots>\SL_n(k\delta). \]

The index records a position in a Cartan flag as well as the degree. Controlling that position is what turns periodicity of real-root chains into the insertion formula of the main theorem.

alphabet order: the chosen total order on the simple-root labels; every theorem on this page allows any such order
standard factorization: $w=L_iR_i$, cut after the longest proper Lyndon prefix
period: the fixed word $W_i$ inserted once whenever the imaginary degree increases by $\delta$
preperiod: the finite real prefix seen before an increasing real-root chain enters its repeating imaginary pattern

Main result

Untwisted affine types

Fix any untwisted affine type of finite rank $n$, any order on its simple-root alphabet, and $1\leq i\leq n$. Write $\SL_i(\delta)=L_iR_i$ as a standard factorization.

Theorem 1.1 · Imaginary insertion There is a cyclic permutation $W_i$ of $\SL_j(\delta)$ for some $j\leq i$ such that, for every $k\geq1$,

\[ \SL_i(k\delta)=L_iW_i^{k-1}R_i,\qquad \SL_i^{\mathrm{ls}}(k\delta)=L_iW_i^{k-1}. \]

The theorem holds for every untwisted type and every alphabet order.

The new all-rank arguments handle the remaining classical types $B$, $C$, and $D$. The exceptional cases use the prior degree-$2\delta$ computations and propagation theorem of Elkins and Tsymbaliuk; this paper does not replace those computations with a new type-independent exceptional proof.

Corollary · Reconstruction from two degrees For a fixed untwisted affine root datum, the words in degrees $\delta$ and $2\delta$ determine the positive system of increasing real-root chains. They recover the finite Dynkin diagram, including bond multiplicities and orientations.

Classical reduction

The finite type-C block grid

When the least letter $r$ has Kac mark two, every degree-$k\delta$ word splits uniquely into $2k$ real blocks. In type $C_n^{(1)}$, the possible block degrees form a $2r$ by $2(n-r)$ rectangle. The grid contains all $4r(n-r)$ positive real roots whose $\alpha_r$-coefficient is one.

$q$ runs across; $p$ runs down. Select any cell.

$d_{0,0}$ has finite projection —.

Its complementary root is —, reflected through the centre because $\delta-d_{p,q}=d_{H-p,J-q}$.

For the displayed alphabet order, its real standard word is

—

Each square is a real block degree. Blue marks the selected root and green marks its complement. The word is the lexicographically greatest shuffle of the two arm prefixes, computed by taking the greater available head.

Exact phase

The period rotates; the preperiod records the cut

Let $B_i$ be the unique Lyndon rotation of $W_i$. There is a unique split $B_i=A_iC_i$ for which $W_i=C_iA_i$, with $0\leq|C_i|<|\delta|$. The increasing-chain preperiod is then $y_i=L_iC_i$, and the real appends are the canonical Lyndon factors of $C_i$.

Figure 3 · Finding the unique rotation cut

the displayed family is $C_3^{(1)}$, order $1<3<0<2$, index $i=3$

candidate suffix length $|C|$ 3

parent $B=A\,C$

rotation $C\,A$

period $W$

preperiod $y=L\,C$

Here $B=123120$, $W=120123$, and the unique cut has $C=120$. Thus $y=L C=10123120$, with $L=10123$. Moving the control compares every possible cyclic cut without changing the theorem's selected period.

Theorem 13.3 · Exact phase and preperiod For every untwisted type, $y_i=L_iC_i$ and $|y_i|\leq2|\delta|-2$. The repeating pattern begins at degree $\delta$ exactly when $C_i$ is empty, and at degree $2\delta$ otherwise.

Companion note

What extends to twisted types

The finite block alphabet and primitive-bracketing results hold for every indecomposable symmetrizable affine root system. Twisted imaginary multiplicities can vary with the degree, so four families cannot use the unchanged degree-$\delta$ indexing of the untwisted theorem.

Proved in $A_2^{(2)}$ With $\delta=\alpha_0+2\alpha_1$, the all-degree formulas for the two alphabet orders are

\[ 0<1:\quad \SL(k\delta)=01(011)^{k-1}1, \]

\[ 1<0:\quad \SL(k\delta)=1(110)^{k-1}10. \]

Established

Finite block alphabets, a nonzero bracketing for primitive zero-weight configurations, the twisted multiplicity obstruction, and the complete $A_2^{(2)}$ formula.

Conjectural

The divisibility-stratified insertion formula in the remaining twisted families. The recorded finite checks support it but do not prove propagation.

Supplementary calculations

Recorded finite checks

The supplied programs construct homogeneous Lie spaces with exact integer or rational arithmetic and select actual costandard brackets before fitting insertion patterns. The main all-rank classical proofs do not depend on these computations.

77twisted type/order cases

214fitted twisted families

129untwisted phase cases

513imaginary-index comparisons

21,952D₄ Jacobi identities

The present page reran two bounded examples: the four $D_4^{(1)}$ words and three $C_3^{(1)}$ words in degree $2\delta$ agreed between the classical constructor and the separate exact checker; both $A_2^{(2)}$ orders were enumerated through degree $6\delta$.

Path	Contents
`manuscript/`	33-page main paper and complete LaTeX source
`extensions/`	8-page companion note on general affine blocks and twisted types
`code/classical/`	all-rank classical constructors, independent exact checks, and recorded outputs
`code/general/`	twisted, folded, triality, model, and phase checks
`SHA256SUMS`	hash manifest for the supplied 55-file editorial package

Scope

What the package does not claim

The main theorem covers every untwisted affine type. Its exceptional cases use cited prior computer-assisted verifications. A type-independent analytic replacement for those exceptional computations is not supplied.

The companion note proves the full insertion formula only for $A_2^{(2)}$ among twisted types. The proposed formulas for the other twisted families remain conjectural; finite calculations establish neither lexicographic maximality in every later flag layer nor the required propagation law.

The package also makes no quantum or shuffle-algebra analogue, categorification, canonical-basis statement, or full Gröbner-Shirshov presentation. The new arguments require coauthor review before submission, and no external submission or referee approval is recorded.

Citation

@misc{kumar2026lyndon,
  title  = {Imaginary standard Lyndon periodicity in untwisted affine types},
  author = {Kumar, Pranava and Gangarapu, Abhijay},
  year   = {2026}
}