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Affine Jacobi-Trudi Identities and $q,t$ -Rogers-Ramanujan Identities

This paper conjectures affine and Hall-Littlewood analogues of dual Jacobi-Trudi identities for orthogonal and symplectic Schur functions indexed by maximal-height rectangular partitions to derive $t$ -analogues of various Rogers-Ramanujan identities, while also proving an affine analogue for rectangular partitions of arbitrary height.

Original authors: S. Ole Warnaar

Published 2026-06-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: S. Ole Warnaar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect trying to build a perfect, symmetrical tower out of blocks. In the world of mathematics, these "blocks" are numbers and shapes called partitions, and the "towers" are complex formulas called identities.

This paper, written by S. Ole Warnaar, is like a new blueprint for building these towers. It connects two very different ways of describing the same structure: one way uses a determinant (a specific grid of numbers that acts like a rigid, pre-fabricated frame), and the other uses a sum (a long list of possibilities that you have to add up).

Here is the story of the paper, broken down into simple concepts:

1. The Old Blueprint: The Jacobi–Trudi Identity

For a long time, mathematicians have known a rule called the Jacobi–Trudi identity. Think of this as a recipe for building a tower of a specific shape (a rectangle).

The Recipe: You take a grid of numbers (a determinant) and calculate it to get the final shape.
The Problem: This recipe works perfectly for simple, standard blocks (called Schur functions). But when you try to build more complex, "twisted" towers (related to symmetries like flipping or rotating, known as orthogonal and symplectic shapes), the old recipe breaks down or doesn't exist.

2. The New Blueprint: Affine Jacobi–Trudi Identities

The author proposes a new set of blueprints (Conjectures 1.2, 1.3, and 1.4) for these complex, twisted towers.

The Innovation: Instead of a simple grid, the new recipe involves an "infinite loop." Imagine a conveyor belt that goes on forever, where you add up contributions from every possible position on the belt, but with a special rule that only certain positions matter.
The "Affine" Twist: The word "Affine" here refers to a specific type of symmetry that repeats itself, like a wallpaper pattern that goes on forever. The author suggests that for these complex towers, you can still use a grid-like formula, but it must account for this infinite, repeating nature.

The Main Claim: The author conjectures (strongly guesses based on evidence) that these new formulas work for specific rectangular shapes. They have proven this works for the simplest case (a single row of blocks) and for a few specific special cases, but the full proof for all cases is still a "work in progress" (a conjecture).

3. The Treasure Hunt: Rogers–Ramanujan Identities

Why does anyone care about these block towers? Because they are the key to unlocking a famous treasure chest: the Rogers–Ramanujan identities.

The Treasure: These are magical equations discovered over a century ago. They say that if you count the ways to build a tower using specific rules (like "no two blocks can be next to each other"), the total number of ways is exactly equal to a simple, elegant product of numbers (like a smooth, flowing river).
The Paper's Contribution: The author uses their new blueprints to find new versions of these magical equations.
- They introduce a "tuning knob" (a variable called $t$ ) that allows them to create a whole family of these equations, not just the original ones.
- They show that these new equations describe the "characters" (the unique signatures) of huge, abstract mathematical structures called Affine Lie Algebras. You can think of these algebras as the "atoms" of symmetry in the universe of math.

4. The "Conditional" Proof

The paper walks a tightrope.

Step 1: The author says, "If my new blueprints (the conjectures) are true, then these new magical equations (Theorems 1.5–1.8) must also be true."
Step 2: The author then says, "But wait! I don't need to wait for the blueprints to be fully proven. I can use a different method (called Ismail's analytic argument) to prove the magical equations are true on their own, without relying on the blueprints."

So, the paper achieves a double victory:

It provides a beautiful, structural explanation (the blueprints) for why these equations exist.
It provides a rigorous, independent proof that the equations are definitely true, even if the blueprints are still being tested.

5. The "Open Problems" (The Unfinished Map)

The paper ends by pointing out parts of the map that are still blank.

The author admits they haven't proven the blueprints for every possible shape yet.
They ask: "Are there other hidden blueprints for different types of symmetries?"
They wonder: "Can we interpret these magical equations as counting colored beads or specific types of patterns?" (This is a hint that there might be a physical or visual way to see these numbers, not just calculate them).

Summary in a Nutshell

This paper is like a mathematician discovering a new, more powerful lens.

Old Lens: Could only see simple, straight towers.
New Lens: Can see complex, twisted, repeating towers.
Result: By looking through this new lens, the author discovers a whole new set of "magic formulas" (Rogers–Ramanujan identities) that connect the shape of these towers to the fundamental laws of symmetry in mathematics. Even though the lens itself is still being perfected (it's a conjecture), the author proves that the magic formulas it reveals are real and correct.

Technical Summary: Affine Jacobi–Trudi Identities and $q, t$ -Rogers–Ramanujan Identities

Problem Statement
The paper addresses the lack of known Jacobi–Trudi identities for significant generalizations of Schur functions, specifically Hall–Littlewood polynomials associated with affine root systems ( $B_n, C_n, BC_n$ ). While classical Jacobi–Trudi and dual Jacobi–Trudi identities express Schur functions as determinants of complete or elementary symmetric functions, analogous formulas for Hall–Littlewood polynomials indexed by rectangular partitions of maximal height have remained largely conjectural. Furthermore, the paper seeks to utilize these identities to derive new $q, t$ -analogues of Rogers–Ramanujan identities for the characters of standard modules of affine Lie algebras.

Methodology
The author employs a combination of symmetric function theory, crystal base theory, and analytic arguments:

Conjectural Framework: The paper proposes affine analogues of the dual Jacobi–Trudi identities for Hall–Littlewood polynomials indexed by rectangular partitions $(k^r)$ . These conjectures express the polynomials as sums over the affine Weyl group (specifically $W(A^{(1)}_{k-1})$ ) of determinants involving elementary symmetric functions with shifted indices and $t$ -dependent weights.
Proof of Theorem 1.1: A rigorous proof is provided for the $GL(n, \mathbb{C})$ case (ordinary Hall–Littlewood polynomials) indexed by rectangular shapes. This proof relies on results from crystal base theory by Schilling and Shimozono regarding level-restricted paths for $U_q(\widehat{\mathfrak{sl}}_k)$ .
Verification of Conjectures: For the $B_n, C_n,$ and $BC_n$ cases, the conjectures are proven for the specific case $k=1$ using bounded Littlewood identities and Bailey pairs (specifically those from Slater's list). Additionally, proofs for $t=1$ are provided for specific identities using cylindric Schur functions.
Derivation of Rogers–Ramanujan Identities: The conjectural Jacobi–Trudi formulas are specialized (setting variables $x_i = q^{i-1/2}$ or $q^i$ and taking the limit $n \to \infty$ ) to derive $q, t$ -Rogers–Ramanujan identities. These derivations utilize Macdonald identities for affine root systems ( $C^{(1)}_k, A^{(2)}_{2k}, D^{(2)}_{k+1}$ ).
Unconditional Proofs: To establish the $q, t$ -Rogers–Ramanujan identities without relying on the unproven Jacobi–Trudi conjectures, the author employs Ismail's analytic argument. This involves showing that the difference between the series and product sides of the identities vanishes at specific points ( $t = q^N$ ) and using the identity theorem for analytic functions.

Key Contributions and Results

Affine Dual Jacobi–Trudi Identities:
- Theorem 1.1: Proves an affine dual Jacobi–Trudi formula for ordinary Hall–Littlewood polynomials $P_{(k^r)}(x; t)$ as a sum over the root lattice $Q$ of determinants involving elementary symmetric functions.
- Conjectures 1.2–1.4: Proposes analogous formulas for $B_n, C_n,$ and $BC_n$ Hall–Littlewood polynomials. These are proven for $k=1$ and for specific $t$ values ( $t=1$ ).
New $q, t$ -Rogers–Ramanujan Identities:
- Theorems 1.5–1.8: Establishes four new families of $q, t$ -Rogers–Ramanujan identities corresponding to the affine root systems $C^{(1)}_k, A^{(2)}_{2k}, D^{(2)}_{k+1},$ and $A^{(2)}_{2k-1}$ . These identities generalize classical results (Rogers–Ramanujan, Andrews–Gordon, GOW identities) by introducing a second parameter $t$ (the Hall–Littlewood parameter) while maintaining a product form on the right-hand side.
- Corollaries 5.1–5.18: Derives a comprehensive list of Rogers–Ramanujan identities for the characters of standard modules of affine Lie algebras ( $A^{(2)}_{2n}, C^{(1)}_n, D^{(2)}_{n+1}, B^{(1)}_n, D^{(1)}_n$ ). This includes eight new identities not previously found in the literature.
Level-Rank Duality: The paper demonstrates a novel manifestation of level-rank duality, showing how specializing the parameter $t$ in the $q, t$ -identities leads to known and new identities for affine Lie algebra characters.
Appendix Results: Proves a new family of Rogers–Ramanujan-type identities (Equation 5.21) related to the $A^{(2)}_{2n}$ case, discovered by Matthew Russell, which connects to the characters of the Virasoro algebra.

Significance
The paper claims to advance the understanding of the combinatorial structure of Hall–Littlewood polynomials by providing the first affine Jacobi–Trudi formulas for rectangular shapes in the $B_n, C_n,$ and $BC_n$ cases (albeit conjecturally for general $k$ ). The primary significance lies in the unification of these combinatorial identities with the representation theory of affine Lie algebras. By deriving $q, t$ -analogues of Rogers–Ramanujan identities, the work extends the classical correspondence between partition identities and affine Lie algebra characters to a two-parameter setting. The unconditional proofs of the resulting Rogers–Ramanujan identities ensure that the new character formulas for standard modules are valid regardless of the status of the underlying Jacobi–Trudi conjectures. The work is dedicated to Jim Lepowsky, acknowledging his foundational role in connecting Rogers–Ramanujan identities to representation theory.