Novel energy preserving bijections between affine crystals for $U_q(\widehat{\mathfrak{sl}}_2)$ and integer partitions

Abstract: Let $B(Λa) \, (a=0,1)$ be the crystal of the level 1 integrable irreducible highest weight representation of the affine quantum group $U_q(\widehat{\mathfrak{sl}}_2)$. We consider the crystal graphs of degree $n$ associated with the irreducible $(2r+1)$-dimensional (resp. $(2r+2)$-dimensional) $U_q(\mathfrak{sl}_2)$ module in $B(Λ_0)$ (resp. $B(Λ_1)$). In this paper, we construct an explicit combinatorial procedure providing a bijection between the set of highest weight paths in these graphs with respect to the action of the Kashiwara operator $\tilde{f}{1}$, and the set of integer partitions of $n$ with sqrank (resp. rerank) $r$, which is a recently introduced partition statistic. As a byproduct, we also obtain a precise interpretation of the motif description of spinons suggested by Bernard-Pasquier-Serban in the spinon picture for Wess-Zumino-Witten conformal field theory models.

• The paper presents explicit energy-preserving bijections that map highest-weight paths in U_q(ˆsl₂) affine crystals to integer partitions characterized by sqrank and rerank.
• It introduces a combinatorial algorithm utilizing bit string encoding, Kashiwara operators, and block replacement to construct and invert the bijections.
• The results establish a direct link between generating functions in representation theory and partition statistics, offering insights into spinon motifs in conformal field theory.

Explicit Energy-Preserving Bijections between Affine Crystals for $U_q(\widehat{\mathfrak{sl}_2})$ and Integer Partitions

Introduction and Objectives

This work constructs new, explicit, energy-preserving bijections between families of affine crystals arising in the representation theory of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}_2})$ and classes of integer partitions characterized by recent partition statistics (sqrank and rerank). The combinatorial nature of these bijections provides a direct link between objects of algebraic combinatorics (crystals and their paths) and partition theory, and makes explicit certain statistical properties and their representation-theoretic interpretations.

The results deepen the connection between the generating functions of integer partition statistics and branching functions of affine Lie algebra modules. They also lead to a combinatorial explanation for the motif description of spinons in Wess-Zumino-Witten (WZW) conformal field theory, as originally suggested by Bernard, Pasquier, and Serban.

Affine Crystals and Partition Statistics

Let $B(\Lambda_a)$, $a=0,1$, denote the crystal of the level-1 integrable irreducible highest weight representation $V(\Lambda_a)$ of $U_q(\widehat{\mathfrak{sl}_2})$. Decomposing $V(\Lambda_a)$ as modules for the subalgebra $U_q(\mathfrak{sl}_2)$ yields, at fixed degree $n$, a connection between the multiplicities of irreducible components and counts of partitions of $n$ with specific values of partition statistics—odd/even minimal excludant [AN2020], or equivalently, sqrank and rerank [Takagi2026].

Explicitly, the number of $U_q(\widehat{\mathfrak{sl}_2})$0-dimensional irreducibles in $U_q(\widehat{\mathfrak{sl}_2})$1 (resp. $U_q(\widehat{\mathfrak{sl}_2})$2-dimensional in $U_q(\widehat{\mathfrak{sl}_2})$3) at degree $U_q(\widehat{\mathfrak{sl}_2})$4 equals the number of partitions of $U_q(\widehat{\mathfrak{sl}_2})$5 with sqrank $U_q(\widehat{\mathfrak{sl}_2})$6 (resp. rerank $U_q(\widehat{\mathfrak{sl}_2})$7).

These statistics, sqrank and rerank, are computed via decompositions of the Young diagram—particularly via the Durfee square/rectangle and rim hook removals—reflecting the combinatorial core of the bijection.

Construction of the Bijections

Kyoto Path Model and Crystal Structure

Using the Kyoto path model [KMN], elements of $U_q(\widehat{\mathfrak{sl}_2})$8 are realized as semi-infinite bit strings (paths) stabilized in the far-left limit to alternating patterns that define the "ground state" of the crystal. The crystal graph is furnished with Kashiwara operators $U_q(\widehat{\mathfrak{sl}_2})$9 and $B(\Lambda_a)$0, with the relevant finite-dimensional $B(\Lambda_a)$1 submodules obtained by considering paths invariant under $B(\Lambda_a)$2 ("$B(\Lambda_a)$3-highest paths").

Bit String Encoding of Partitions

Given an integer partition $B(\Lambda_a)$4 (with associated Young diagram), the bijection proceeds through several steps:

1. Restriction Step: Decompose $B(\Lambda_a)$5 into Durfee piece $B(\Lambda_a)$6 and a restricted subdiagram $B(\Lambda_a)$7 via rim hook removals determined by the arm lengths correlated with the relevant partition statistic.
2. Bit String Map: $B(\Lambda_a)$8 is encoded as a finite bit string of fixed length with prescribed numbers of $B(\Lambda_a)$9s and $a=0,1$0s, directly corresponding to the multiplicity structure of the crystal tensor powers.
3. Application of Kashiwara Operators: The bit string is promoted to a $a=0,1$1-highest path via iterative application of $a=0,1$2 (raising) operators.
4. Lengthening via Block Replacement: Consecutive $a=0,1$3 pairs are systematically expanded to $a=0,1$4 blocks, producing a longer bit string that encodes additional combinatorial data and sets up a direct link between energy gradings and cell counts in the partition.
5. Block Insertion and Energy Calculation: Specific insertion rules for $a=0,1$5-blocks and $a=0,1$6-blocks at "spots" determined by the string decomposition of the bitstring enable additional combinatorial operations, maintaining energy conservation. The total energy (as defined on the path, mirroring the excitation degree in the crystal) exactly matches the size $a=0,1$7 of the original partition.

The process is invertible: the structure of the resulting path allows unique recovery of the original partition, establishing a bijection.

Properties and Strong Results

• Energy Preservation: The constructed bijections are energy-preserving; the excitation level of the path in the crystal is equal to the integer partition size $a=0,1$8. This is a nontrivial matching that goes beyond cardinality or dimension count and aligns entire graded generating functions.
• Dimensional Correspondence: The bijection precisely matches $a=0,1$9- and $V(\Lambda_a)$0-dimensional $V(\Lambda_a)$1 submodules within a given degree with partitions of fixed sqrank or rerank.
• Invertibility: Each step in the construction is algorithmically reversible, yielding explicit inverse bijections.

Implications for Physics and Representation Theory

Spinon Motif Interpretation

The combinatorics of bit strings and paths also realize a physical interpretation in the context of WZW conformal field theory. In the motif construction of spinons [BPS1994, BLS1994], the bit string decomposes into "strings" containing two spinons each, with precise charge assignments recoverable from the local bit pattern. The total energy (path degree) then matches the Virasoro $V(\Lambda_a)$2-grading in the conformal field theory spinon basis—directly tying the combinatorial description to spectrum-generating algebraic structures.

Character and Branching Function Connections

The explicit combinatorial bijection gives a constructive explanation for, and strengthens, previously known generating function coincidences:

• Branching functions from the Weyl-Kac formula for $V(\Lambda_a)$3,
• Partition generating functions stratified by minimal excludant-type statistics,
• Configuration sum representations in solvable lattice models and box-ball systems [CKST2023].

This combinatorial matching may be refined or generalized to other types of affine algebras or root systems, indicating a broader framework for correspondence between representation-theoretic structures and partition statistics.

Theoretical and Practical Developments

• Algorithmic Representation: The bijections provide effective algorithms for enumerating highest weight paths by combinatorial partition methods and vice versa, which may facilitate statistical sampling or enumeration in both algebraic combinatorics and mathematical physics simulations.
• Generalization Potential: The techniques point toward possible generalizations to higher rank algebras, colored partitions, or different kinds of crystals, suggesting a rich avenue for future combinatorial and representation-theoretic research.
• Physical Spectrum Analysis: The matched gradings and motif interpretation strengthen the use of combinatorial bases (crystals/paths) for analyzing spectra in integrable field theories and related solvable models.

Future Directions

• Extensions to Higher Rank or Other Affine Types: Can analogous combinatorial and energy-preserving bijections be constructed for crystals of $V(\Lambda_a)$4 or other quantum affine algebras?
• Relation to Fermionic/Bosonic Character Formulas: How does the explicit bijection structure clarify the equivalence of various fermionic/bosonic character representations and their connections to configuration sums?
• Refinement of Partition Statistics: Are there new partition statistics (or families thereof) whose generating functions match other branching functions or crystal decompositions beyond the minimal excludant, sqrank, and rerank?
• Further Physical Applications: Investigation into new spinon-type or motif-related combinatorics for other algebraic and physical systems, particularly those described by crystal base theory.

Conclusion

The paper presents an explicit, combinatorial algorithm for constructing energy-preserving bijections between specific highest-weight paths in affine crystals for $V(\Lambda_a)$5 and integer partitions equipped with novel, refined statistics (sqrank/rerank). This provides a direct, constructive explanation for the identity of certain partition-counting functions and representation-theoretic branching functions, sheds light on the combinatorial content of physical motifs in conformal field theory, and opens a pathway to further algorithmic and theoretical developments linking combinatorics and quantum algebra.

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References:

• Takagi, T., "A Polynomial Bosonic Form of Statistical Configuration Sums and the Odd/Even Minimal Excludant in Integer Partitions," Annals of Combinatorics (2026) (2605.30806).
• Kang et al., "Affine crystals and vertex models," Int. J. Mod. Phys. A (1992).
• Bernard, D., Pasquier, V., Serban, D., "Spinons in Conformal Field Theory," Nucl. Phys. B428 (1994).
• Andrews, G.E., Newman, D., "The minimal excludant in integer partitions," J. Integer Sequences (2020).