Verma Bases for finite dimensional Representations of the orthosymplectic Lie superalgebra $\mathfrak{spo}(4|1)$

Abstract: We define the Verma vector system for each finite dimensional irreducible representation of the orthosymplectic Lie superalgebra $\mathfrak{spo}(4|1)$ with the highest weight $λ,$ via the conditions that making a tableau with shape $λ$ to be a Kashiwara-Nakashima tableau. We then show the linearly independence of this vector system. It turns out to be a basis of the finite dimensional irreducible representation $L(λ)$ of the orthosymplectic Lie superalgebra $\mathfrak{spo}(4|1)$ with the highest weight $λ,$ which analogs to the Verma basis of representations of $\mathfrak{sp}_4,$ called the Verma basis of the finite dimensional irreducible representation of $\mathfrak{spo}(4|1)$.

• The paper constructs explicit Verma bases for finite dimensional irreducible representations of the orthosymplectic Lie superalgebra $\mathfrak{spo}(4|1)$ using a weight-preserving bijection between Verma vectors and Kashiwara–Nakashima tableaux.
• Exponent inequalities constrain the structure of Verma bases, derived from Lie algebra root lengths and combinatoric constraints.
• The Verma bases provide a constructive understanding of representations and internal structure.

Overview

This paper constructs explicit monomial bases — Verma bases — for all finite dimensional irreducible representations of the orthosymplectic Lie superalgebra $\mathfrak{spo}(4|1)$. The construction extends the authors' earlier work on $\mathfrak{sp}_4$, where a one-to-one correspondence between Verma vectors and Kashiwara–Nakashima (KN) tableaux was established. The central observation is that the inequalities governing the exponents of the negative simple root vectors in the Lie algebra case coincide with the combinatorial conditions characterizing KN tableaux; the same phenomenon persists in the superalgebra setting, where the odd simple root introduces additional parity constraints. The main result has two parts: a weight-preserving bijection between the set $H$ of Verma vectors and the set $\mathrm{KN}_{\lambda}(4|1)$ of KN tableaux of shape $\lambda$, and a proof that $H$ is a basis of $L(\lambda)$.

The algebra and its representations

The paper works with the matrix realization of $\mathfrak{spo}(4|1)$ as a $5\times 5$ matrix superalgebra whose even part is isomorphic to $\mathfrak{sp}_4$. With Cartan subalgebra spanned by $\mathfrak{sp}_4$0 and $\mathfrak{sp}_4$1, the root system decomposes into even roots $\mathfrak{sp}_4$2 and odd roots $\mathfrak{sp}_4$3. The chosen simple system $\mathfrak{sp}_4$4 contains one even and one odd simple root, connected by a double bond in the Dynkin diagram. Finite dimensional irreducibles are parametrized by highest weights $\mathfrak{sp}_4$5 with $\mathfrak{sp}_4$6, equivalently partitions $\mathfrak{sp}_4$7 with at most two parts. This integrability criterion follows from Kac's classification and Shader's work on $\mathfrak{sp}_4$8.

The negative simple root vectors are $\mathfrak{sp}_4$9 (even, weight $H$0) and $H$1 (odd, weight $H$2). A Verma vector is a monomial of the PBW-ordered form

$H$3

with exponents constrained by the system

$H$4

Note that the bound on $H$5 involves a minimum of two terms — an artifact absent from the pure symplectic case — reflecting the interplay between the odd root string length and the even root structure. The floor-function bound on $H$6 arises because $H$7 is odd: applying $H$8 twice to a vector can annihilate it or produce sign cancellations tied to the presence of the entry $H$9 in the corresponding tableau.

Kashiwara–Nakashima tableaux and the exponent inequalities

KN tableaux for $\mathrm{KN}_{\lambda}(4|1)$0 use the alphabet $\mathrm{KN}_{\lambda}(4|1)$1 with ordering $\mathrm{KN}_{\lambda}(4|1)$2, subject to row/column monotonicity conditions (with $\mathrm{KN}_{\lambda}(4|1)$3 not repeating along rows but allowed to repeat down columns), exclusion of $\mathrm{KN}_{\lambda}(4|1)$4 and $\mathrm{KN}_{\lambda}(4|1)$5 in the same column, and two forbidden adjacent-column configurations. Following Liu–Yang's crystal-theoretic treatment of $\mathrm{KN}_{\lambda}(4|1)$6, the paper invokes the fact that KN tableaux label the crystal basis of the quantum deformation $\mathrm{KN}_{\lambda}(4|1)$7 to conclude that $\mathrm{KN}_{\lambda}(4|1)$8. This dimension count is essential: combined with the bijection and the independence proof, it upgrades linear independence to a basis statement without requiring an independent spanning argument.

For each tableau $\mathrm{KN}_{\lambda}(4|1)$9, four statistics $\lambda$0 are defined by counting entries in each row relative to thresholds ($\lambda$1, $\lambda$2, $\lambda$3, etc.). The key structural lemma is that these statistics automatically satisfy exactly the inequality system above, with the minima and floors emerging from case analysis over the possible placements of $\lambda$4 and the forbidden column patterns. Conversely, every quadruple satisfying the inequalities determines a unique KN tableau via explicit templates covering seventeen cases, distinguished by the parities of $\lambda$5 and $\lambda$6 and by boundary equalities such as $\lambda$7 and $\lambda$8. The bijection is verified to be weight-preserving: direct computation gives $\lambda$9 uniformly across all cases, matching the weight of the monomial computed from the root data. An illustrative example with $H$0 enumerates the full correspondence.

Linear independence

The independence argument adapts Raghavan–Sankaran's method for $H$1. The module $H$2 is realized inside $H$3, where $H$4 denotes the super exterior product (exterior on the even part of the natural module $H$5, symmetric on the odd line), and $H$6 is a maximal vector. Column-strict Young tableaux (KN conditions weakened) index a subset of the standard monomial basis of $H$7, hence are linearly independent.

A total order on column-strict tableaux is introduced, reading entries from rightmost column downward. The technical core is an explicit description of $H$8: using operators $H$9 acting on tensor factors, which anticommute for distinct factors, one obtains $L(\lambda)$0 and closed formulas showing that $L(\lambda)$1 expands as a positive multiple of its leading tableau plus strictly smaller terms, with coefficient $L(\lambda)$2. Iterating through the four factors of the monomial — $L(\lambda)$3, then $L(\lambda)$4, then $L(\lambda)$5, then $L(\lambda)$6 — yields the triangular expansion

$L(\lambda)$7

with $L(\lambda)$8. Triangularity with respect to the total order, together with independence of the $L(\lambda)$9, forces any vanishing linear combination of Verma vectors to have all coefficients zero by a descending induction on the largest tableau appearing. Since $\mathfrak{spo}(4|1)$0, the Verma vectors form a basis.

Two features of this proof deserve emphasis. First, the positivity of the leading coefficients depends on the factorial structure of the $\mathfrak{spo}(4|1)$1-action on the symmetric (odd) wedge factors; the anticommutation relation $\mathfrak{spo}(4|1)$2 is what makes the square collapse to a sum of single-factor squares. Second, the argument relies on the realization of $\mathfrak{spo}(4|1)$3 as a submodule of $\mathfrak{spo}(4|1)$4 generated by $\mathfrak{spo}(4|1)$5; the paper does not reprove that this submodule is irreducible, citing instead the standard model for these representations.

Limitations and open questions

The result is confined to rank two: both the seventeen-case tableau classification and the case-by-case weight verification depend on the specific structure of $\mathfrak{spo}(4|1)$6, and no generalization to $\mathfrak{spo}(4|1)$7 for $\mathfrak{spo}(4|1)$8 is attempted. The proof that the statistics $\mathfrak{spo}(4|1)$9 satisfy the inequality system proceeds by exhaustive case analysis rather than a uniform argument, so extending the method requires redoing this analysis type by type. Additionally, the dimension formula is imported from the quantum group crystal theory of Liu–Yang rather than proved directly at $5\times 5$0; a self-contained combinatorial proof of $5\times 5$1 would make the basis theorem independent of crystal basis machinery. Whether the Verma basis constructed here admits structural refinements — for instance, whether it coincides with or relates to a canonical or global crystal basis of $5\times 5$2 under specialization — is left unaddressed.

Conclusion

The paper establishes that finite dimensional irreducible $5\times 5$3-modules admit Verma bases indexed by Kashiwara–Nakashima tableaux, generalizing the $5\times 5$4 construction of the same authors. The three ingredients — the inequality system defining admissible exponent tuples, the weight-preserving bijection with KN tableaux, and the triangularity-based independence proof — combine to give a complete, constructive description of these bases. The work provides a concrete foundation for branching and character computations in this superalgebra, while leaving the extension to higher-rank orthosymplectic superalgebras as the natural open problem.