Irreducibility criterion for tensor products of a snake module with an arbitrary simple module over quantum affine $\mathfrak{sl}_k$ (Bittmann–Li Conjectures 5.10 and 5.6)

For $q$ not a root of unity, it is a basic open problem to decide when the tensor product $L(M)\otimes L(N)$ of two simple finite-dimensional modules over the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_k)$ is again simple. Bittmann and Li (2022) combined Lapid–Mínguez's criterion for parabolic induction of $p$-adic $GL_N$ with quantum affine Schur–Weyl duality and an explicit Grothendieck-ring isomorphism to prove such a criterion for every rank $k\geq 1$ in two cases: both factors snake modules, or one factor a fundamental module at an extremal node and the other arbitrary. In both cases irreducibility is equivalent to two explicit matching conditions $\mathrm{LC}_k(m_M,m_N)$ and $\mathrm{LC}_k(m_N,m_M)$ on the associated multisegments. They conjecture the same criterion for the asymmetric case of an arbitrary snake module tensored with an arbitrary simple module (Conjecture 5.10), with the identification of the matching conditions with socle/cosocle conditions for arbitrary ladders (Conjecture 5.6) as the expected key step. The conjectured criterion, and with it any effective irreducibility criterion for general tensor products in this category, remains open.

Background

Let $q\in\mathbb{C}^{\times}$ be not a root of unity and let $U_q(\widehat{\mathfrak{sl}}_k)$ be the quantum affine algebra of type $A_{k-1}$, the quantized enveloping algebra of the loop algebra of $\mathfrak{sl}_k$. The category $\mathscr{C}_k$ of finite-dimensional type-1 representations — the standard class of finite-dimensional modules, containing all simple modules, on which the loop parameter enters only through integral powers — is a non-semisimple braided tensor category whose simple objects $L(M)$ are classified, by work of Chari and Pressley, by dominant monomials $M$ in commuting variables $Y_{i,r}$ ($1\leq i\leq k-1$, $r\in\mathbb{Z}$). Its Grothendieck ring is encoded by the $q$-character map $\chi_q$ of Frenkel–Reshetikhin, an injective ring homomorphism from $K_0(\mathscr{C}_k)$ to a Laurent polynomial ring in the $Y_{i,r}^{\pm 1}$; the algorithm of Frenkel–Mukhin computes $q$-characters of many simple modules iteratively. A structural question open since the 1990s is: for which pairs of simple modules is $L(M)\otimes L(N)$ simple? This also matters for the cluster-algebra structure of the Grothendieck ring discovered by Hernandez–Leclerc (a cluster algebra being a commutative ring equipped with a distinguished set of generators, its cluster variables, produced by an iterative mutation process): a simple module is called real when its tensor square is simple, and compatibility of cluster variables is expected to correspond to simplicity of the corresponding tensor product.

A parallel combinatorial theory exists for representations of $GL_N(F)$ over a non-archimedean field $F$. Irreducibles are classified by Zelevinsky multisegments, finite formal sums of segments $\Delta=[a,b]$ with $a\leq b$; $Z(m)$ denotes the irreducible attached to $m$ by the Zelevinsky classification, and the parabolic induction of two irreducibles — the representation of $GL_{N+M}(F)$ obtained by inducing $Z(m)\otimes Z(n)$ up from the Levi factor $GL_N(F)\times GL_M(F)$, where $Z(m)$ and $Z(n)$ are representations of $GL_N(F)$ and $GL_M(F)$ — is written $Z(m)\times Z(n)$. A multisegment is a ladder if both endpoints of its segments increase along the sum; $\Delta\prec\Delta'$ means $b(\Delta)+1\leq b(\Delta')\leq e(\Delta)+1\leq e(\Delta')$. Lapid–Mínguez (2016) gave a purely combinatorial criterion for irreducibility of $Z(m)\times Z(n)$ with $m$ a ladder and $n$ arbitrary, in terms of matching functions between the sets $X_{m,n}=\{(i,j):\Delta_i\prec\Delta'_j\}$ and $Y_{m,n}=\{(i,j):\overleftarrow{\Delta_i}\prec\Delta'_j\}$ of linked segment pairs, where $\overleftarrow{\Delta}=[a-1,b-1]$ denotes the segment $\Delta=[a,b]$ shifted down by one (the condition $\mathrm{LC}(m,n)$); their 2018 paper connected these conditions to the stronger notion of $\square$-irreducibility (a representation is $\square$-irreducible when its parabolic induction with any irreducible representation is again irreducible).

The two worlds are bridged by quantum affine Schur–Weyl duality (Jimbo; Chari–Pressley): a monoidal functor from modules over the affine Hecke algebra $\widehat{\mathcal{H}}_N(q^2)$, which by type theory for $p$-adic groups (Bushnell–Kutzko, Heiermann) realizes Bernstein blocks of $GL_N(F)$, to the level-$N$ subcategory of $\mathscr{C}_k$. The functor is an equivalence of categories only when $N<k$, and it sends a segment $[a,b]$ to $L(Y_{b-a+1,-a-b})$; ladders correspond exactly to snake modules, the simple modules introduced by Mukhin–Young whose highest weights form a staircase chain of the $Y_{i,r}$. Hence Lapid–Mínguez's criterion transfers to tensor products of a snake module with an arbitrary simple module only in the large-$k$ regime.

Bittmann–Li (2022) treat all $k\geq 1$. They construct a category $\mathcal{C}_k$ by quotienting and localizing the $p$-adic-side category (segments of length $>k$ are sent to zero, segments of length $k$ to the trivial representation), through which the Schur–Weyl functor factors as an exact monoidal functor $\widetilde{\mathcal{F}}:\mathcal{C}_k\to\mathscr{C}_k$, and they prove that $\widetilde{\mathcal{F}}$ induces an isomorphism of Grothendieck rings. Writing $\Delta\prec_k\Delta'$ for $\Delta\prec\Delta'$ with $e(\Delta')-b(\Delta)<k$, they define modified matching conditions $\mathrm{LC}_k(m,n)$ using the sets $X^{(k)}_{m,n}$, $Y^{(k)}_{m,n}$ built from $\prec_k$ instead of $\prec$, and analogously $\mathrm{RC}_k$. Their main theorem states that for two snake modules, $L(M)\otimes L(M')$ is simple if and only if both $\mathrm{LC}_k(m_M,m_{M'})$ and $\mathrm{LC}_k(m_{M'},m_M)$ hold; the same is proved when one factor is a fundamental module $L(Y_{1,r})$ or $L(Y_{k-1,r})$ at an extremal node of the Dynkin diagram and the other is an arbitrary simple module. Via the Hernandez–Leclerc correspondence and the parametrization of the dual canonical basis of the quotient cluster algebra $\mathbb{C}[\mathrm{Gr}(k,n,\sim)]$ by rectangular semistandard Young tableaux (Chang–Duan–Fraser–Li), this also yields a tableau-compatibility criterion generalizing the classical result of Leclerc–Zelevinsky that two Plücker coordinates — the standard generators of the Grassmannian coordinate ring, indexed by $k$-element column sets — are compatible if and only if their column sets are weakly separated.

What the paper leaves open is the asymmetric case: one factor an arbitrary snake module, the other an arbitrary simple module (Conjecture 5.10 in the arXiv v3 numbering; Conjecture 6.13 in the v2 numbering). The expected key ingredient, that $\mathrm{LC}_k$ and $\mathrm{RC}_k$ identify with the socle and cosocle conditions $\mathrm{LI}_k$ and $\mathrm{RI}_k$ (socle and cosocle meaning maximal semisimple submodule and maximal semisimple quotient; $\mathrm{LI}_k(\pi,\sigma)$ says $Z(m+n)=\mathrm{soc}(\pi\times\sigma)$ in $\mathcal{C}_k$) for arbitrary ladders and not just singleton segments $[a,a]$, is likewise conjectural (Conjecture 5.6, v2 Conjecture 6.8). The paper additionally records two further open statements: that $\widetilde{\mathcal{F}}$ is an equivalence of categories (Conjecture 3.7, v2 Conjecture 3.6), and that the tableau criterion should lift from the quotient ring $\mathbb{C}[\mathrm{Gr}(k,n,\sim)]$ to the full homogeneous coordinate ring $\mathbb{C}[\mathrm{Gr}(k,n)]$ (Remark 7.9, v2 Remark 6.14); these are adjacent problems with their own objectives. The fully general problem — an effective criterion for $L(M)\otimes L(N)$ with both factors arbitrary — remains open in the field at large.

Problem Statement

Prove or refute the conjectured irreducibility criterion of Bittmann–Li for tensor products with one snake factor, valid for every rank $k\geq 1$:

(1) Conjecture 5.10 (arXiv v3 numbering; Conjecture 6.13 in v2). For every $k\geq 1$, every simple snake module $L(M)$ of $U_q(\widehat{\mathfrak{sl}}_k)$ — that is, every simple module whose multisegment $m_M$, read off the dominant monomial via $[a,b]\mapsto Y_{b-a+1,-a-b}$, is a ladder — and every arbitrary simple module $L(N)$: the tensor product $L(M)\otimes L(N)$ is irreducible if and only if the two matching conditions $\mathrm{LC}_k(m_M,m_N)$ and $\mathrm{LC}_k(m_N,m_M)$ both hold. Here the sets of $k$-linked segment pairs are $X^{(k)}_{m,n}=\{(i,j):\Delta_i\prec_k\Delta'_j\}$ and $Y^{(k)}_{m,n}=\{(i,j):\overleftarrow{\Delta_i}\prec_k\Delta'_j\}$, where, for segments $\Delta=[b(\Delta),e(\Delta)]$, $\Delta\prec\Delta'$ means $b(\Delta)+1\leq b(\Delta')\leq e(\Delta)+1\leq e(\Delta')$, $\Delta\prec_k\Delta'$ means $\Delta\prec\Delta'$ and $e(\Delta')-b(\Delta)<k$, and $\overleftarrow{\Delta}=[b(\Delta)-1,e(\Delta)-1]$ is the shifted segment; each condition $\mathrm{LC}_k(m,n)$ requires an injective map $f\colon X^{(k)}_{m,n}\to Y^{(k)}_{m,n}$, $f(i,j)=(i',j')$, such that for every $(i,j)\in X^{(k)}_{m,n}$ either $i=i'$ and $\Delta'_{j'}\prec_k\Delta'_j$, or $j=j'$ and $\Delta_i\prec_k\Delta_{i'}$.

(2) The conjectured key ingredient, Conjecture 5.6: for any ladder multisegment $m\in\mathrm{Mult}_k$ and any multisegment $n\in\mathrm{Mult}_k$, the condition $\mathrm{LC}_k(m,n)$ is equivalent to the socle condition $\mathrm{LI}_k(Z(m),Z(n))$, and $\mathrm{RC}_k(m,n)$ to the cosocle condition $\mathrm{RI}_k(Z(m),Z(n))$ — currently proved only for singleton segments $m=[a,a]$ (Proposition 5.5).

A complete answer establishes (1) for all $k\geq 1$ and all snake/arbitrary pairs, either directly or by first establishing (2) and deducing (1); the definitions of $\mathcal{C}_k$, $\prec_k$, the sets $X^{(k)},Y^{(k)}$, matching functions, and ladders are those of the source paper. A single explicit counterexample — a specific $k\geq 1$, an explicit snake dominant monomial $M$ and dominant monomial $N$, with the composition factors of $L(M)\otimes L(N)$ computed, exhibiting a mismatch between irreducibility and the two $\mathrm{LC}_k$ conditions — refutes the conjecture and equally resolves the question. The surrounding general problem (an effective criterion for two arbitrary simple factors) and the paper's separate open statements — the equivalence of categories $\widetilde{\mathcal{F}}$ (Conjecture 3.7) and the lift of the tableau criterion to $\mathbb{C}[\mathrm{Gr}(k,n)]$ (Remark 7.9) — are context and are not required for completion.

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