Simplicity of tensor products of Kirillov-Reshetikhin modules for the non-nested interval collections of Kanakubo-Koshevoy-Nakashima (Conjecture 7.11)

For an untwisted quantum affine algebra of simply-laced ADE type, Kanakubo, Koshevoy and Nakashima proved that the q-characters of Kirillov-Reshetikhin (KR) modules whose supports form a nested collection of integer intervals appear as cluster variables of seeds produced by maximal green sequences of the Hernandez-Leclerc cluster algebra, and that the corresponding tensor products are simple; for five explicit families of non-nested interval collections they constructed such green sequences as well, and conjectured that the tensor products are again simple (Conjecture 7.11 of their paper, the subject of this problem). The task is to prove or disprove this conjecture: for each of the types $A_n^{(1)}$ , $D_n^{(1)}$ , $E_6^{(1)}$ , $E_7^{(1)}$ , $E_8^{(1)}$ and every admissible choice of the integer parameters $m$ and $l_1<\cdots<l_n$ , decide whether the tensor product of the KR modules attached to the corresponding non-nested collection is a simple module. By a theorem of Hernandez this is equivalent to every pair of modules in the collection strongly commuting. The conjecture is open: Oh and Scrimshaw determined the R-matrix denominator formulas between two arbitrary KR modules, and with them the simplicity of all two-fold KR tensor products, in the nonexceptional types, which makes the type $A$ and type $D$ instances reducible to an as-yet unpublished uniform verification, while for the exceptional types $E_6$ , $E_7$ , $E_8$ the required uniform denominator formula is itself conjectural and the i-box techniques that settle the nested case provably do not apply.

Background

Let $\mathfrak g$ be a complex simple Lie algebra of simply-laced type $A_n$ ( $n\ge1$ ), $D_n$ ( $n\ge4$ ), $E_6$ , $E_7$ or $E_8$ (the ADE types), with Coxeter number $h$ (the order of a Coxeter element of the Weyl group: $h=n+1$ , $2n-2$ , $12$ , $18$ , $30$ in the five respective series), and let $U_q(\widehat{\mathfrak g})$ be the associated untwisted quantum affine algebra, the quantized enveloping algebra of the affine Kac-Moody algebra. The category $\mathcal C$ of finite-dimensional $U_q(\widehat{\mathfrak g})$ -modules is a monoidal category which is not semisimple: the tensor product of two simple modules need not be simple, and characterizing exactly which tensor products remain simple is a central structural problem.

Frenkel and Reshetikhin introduced the $q$ -character of a finite-dimensional $U_q(\widehat{\mathfrak g})$ -module, an injective ring homomorphism from the Grothendieck ring $K_0(\mathcal C)$ into a ring of Laurent polynomials in variables $Y_{v,a}$ indexed by a Dynkin node $v$ and a spectral parameter $a\in\mathbb C^\times$ ; it refines the ordinary character by splitting weight spaces according to the loop rotation. The simple modules are parametrized by their dominant highest monomials, a monomial being dominant when all of its exponents are non-negative. In the shifted variables $t_{v,k}$ ( $k\in\mathbb Z_{\ge1}$ ) used by the source paper, related to the $Y$ -variables by $t_{v,k}\leftrightarrow Y_{v,q^{-2k+2}}$ when $v$ is a source vertex of the Dynkin quiver with alternating orientation (each vertex of the Dynkin diagram is made either a source, all arrows at it pointing away from it, or a sink, all arrows at it pointing into it) and $t_{w,k}\leftrightarrow Y_{w,q^{-2k+1}}$ when $w$ is a sink, a dominant monomial $M=\prod_{j=1}^{k}t_{v,r+j}$ defines the Kirillov-Reshetikhin (KR) module $W^{(v)}_{k,r+k}=V(M)$ , and the integer interval $[r+1,r+k]$ is called its support; for $k=1$ these are the fundamental modules. The Frenkel-Mukhin algorithm computes q-characters from dominant monomials.

Simplicity of tensor products is controlled by R-matrices and by a pairwise criterion. For simple modules $M,N$ the normalized R-matrix gives a homomorphism $M\otimes N\to N\otimes M$ ; Kang-Kashiwara-Kim-Oh proved that if one factor is real (its self-tensor product is simple), then $M\otimes N$ is simple if and only if this R-matrix is an isomorphism, which in turn is governed by the denominator formula: a Laurent polynomial $d_{M,N}(z)$ in the ratio $z$ of the two spectral parameters whose roots are exactly the poles of the R-matrix. Independently of realness, Hernandez proved that a tensor product $S_1\otimes\cdots\otimes S_N$ of simple modules is simple if and only if $S_i\otimes S_j$ is simple for every pair $i<j$ ; pairs satisfying the latter are said to strongly commute. Hence a tensor product of KR modules is simple exactly when all of its pairs strongly commute, and the question is pairwise.

On the cluster-algebra side: a cluster algebra is the commutative ring generated by cluster variables obtained from an initial seed (a quiver with one variable per vertex) by iterated mutations. Hernandez and Leclerc attached to $\mathfrak g$ the infinite quiver $Q\boxtimes A_\infty$ , where $Q$ is the Dynkin quiver with alternating orientation, and proved that after a monomial change of variables certain cluster variables coincide with the q-characters of KR modules whose supports sit at level at least $h/2$ ; for a module $W^{(v)}_{a,L}$ of length $a$ with right end $L$ (the source's notation) with $L\le a+h/2$ the un-shifted construction yields only truncated q-characters, computed inside the truncated subcategories $\mathcal C_\ell$ of modules whose spectral parameters are constrained to a bounded window, unless one first shifts the variables. A maximal green sequence of a quiver is a finite sequence of mutations at green vertices (vertices receiving no arrow from the frozen part of the framed quiver, the quiver with one frozen copy appended to each mutable vertex) that turns every mutable vertex red; it produces a distinguished terminal seed and a cluster Donaldson-Thomas transformation, a birational map comparing initial and terminal cluster variables.

Kanakubo-Koshevoy-Nakashima proved that for a nested collection of intervals (any two intervals of the collection are nested, one contained in the other) there is a green sequence whose terminal seed contains the q-characters of the KR modules of the collection among its cluster variables (their Theorem 7.4), and that the tensor product of these modules is simple in any order (their Theorem 7.5); the proof goes through the admissible-chain and $i$ -box theory of Kashiwara-Kim-Oh-Park - an $i$ -box being a pair (Dynkin node, interval) to which a KR module is attached in a normalization where their strong-commutation theorems apply - combined with Hernandez's criterion. For the non-nested collections $\mathbf C_\cdot(l_1,\dots,l_n;m)$ defined in the problem statement they likewise constructed green sequences with the q-character property (their Theorem 7.10) and conjectured simplicity (their Conjecture 7.11). Their Remark 7.12 records the obstruction: intervals with right endpoints differing by two do not give strongly commuting pairs, and the collections of the conjecture do not correspond to chains of commuting $i$ -boxes, so the method of Theorem 7.5 does not extend. The strongest adjacent results are those of Oh and Scrimshaw, who determined the denominator formulas between two arbitrary KR modules and, as a consequence, completely classified simplicity of tensor products of KR modules in all nonexceptional affine types; for the simply-laced exceptional types $E_{6,7,8}^{(1)}$ the uniform normal form of the KR denominator formula is conjectural in their work. No published work applies these formulas to the collections of Theorem 7.10 or addresses Conjecture 7.11.

Problem Statement

Fix a simply-laced type among $A_n^{(1)}$ ( $n\ge1$ ), $D_n^{(1)}$ ( $n\ge4$ ), $E_6^{(1)}$ , $E_7^{(1)}$ , $E_8^{(1)}$ , with Coxeter number $h$ (respectively $n+1$ , $2n-2$ , $12$ , $18$ , $30$ ). Number the vertices of the Dynkin diagram by the standard Bourbaki (Sage) numbering used by the source paper: for $A_n$ , the path $1-2-\cdots-n$ ; for $D_n$ , the path $1-2-\cdots-(n-2)$ with the two additional vertices $n-1$ and $n$ both attached to $n-2$ ; for $E_6$ , $E_7$ , $E_8$ , the path $1-3-4-5-6-7-8$ (truncated after $6$ in type $E_6$ and after $7$ in type $E_7$ ) with the additional vertex $2$ attached to $4$ . Let $Q$ be this Dynkin diagram equipped with the alternating orientation used by the source, in which every vertex is either a source (all arrows at it point away from it) or a sink (all arrows at it point into it): in types $A_n$ and $D_n$ the sources are the even-labeled vertices of the path; in type $D_n$ the two attached vertices $n-1$ and $n$ are sources when $n$ is odd and sinks when $n$ is even; in types $E_6$ , $E_7$ , $E_8$ the sources are the vertices $2$ , $3$ , $5$ , $7$ and the sinks are the vertices $1$ , $4$ , $6$ , $8$ (as far as these labels exist in the type). Write $v_i$ for the vertex labeled $i$ , so the vertices of $Q$ are $v_1,\ldots,v_n$ .

Let $m$ and $l_1<\cdots<l_n$ be integers with $m\ge h/2+1$ and $l_i\ge h/2$ for all $i$ , such that every interval below is non-empty (its right end is at least its left end). Define the collection of intervals $\mathbf C=\mathbf C_\cdot(l_1,\dots,l_n;m)$ by

\mathbf C_{A_n}(l_1,\dots,l_n;m)=\{[\,l_i,\;m+\lfloor i/2\rfloor+1\,]\mid i=1,\dots,n\},

\mathbf C_{D_n}(l_1,\dots,l_n;m)=\{[\,l_i,\;m+\lfloor i/2\rfloor+1\,]\mid i=1,\dots,n-2\}\cup\{[\,l_{n-1},\;m+n/2-1\,],[\,l_n,\;m+n/2-1\,]\}\quad(n\ \mathrm{even}),

\mathbf C_{D_n}(l_1,\dots,l_n;m)=\{[\,l_i,\;m+\lfloor i/2\rfloor+1\,]\mid i=1,\dots,n-2\}\cup\{[\,l_{n-1},\;m+(n-1)/2\,],[\,l_n,\;m+(n-1)/2\,]\}\quad(n\ \mathrm{odd}),

\mathbf C_{E_6}(l_1,\dots,l_6;m)=\{[\,l_1,m\,],[\,l_2,m+1\,],[\,l_3,m+1\,],[\,l_4,m+1\,],[\,l_5,m+2\,],[\,l_6,m+2\,]\},

\mathbf C_{E_7}(l_1,\dots,l_7;m)=\{[\,l_1,m\,],[\,l_2,m+1\,],[\,l_3,m+1\,],[\,l_4,m+1\,],[\,l_5,m+2\,],[\,l_6,m+2\,],[\,l_7,m+3\,]\},

\mathbf C_{E_8}(l_1,\dots,l_8;m)=\{[\,l_1,m\,],[\,l_2,m+1\,],[\,l_3,m+1\,],[\,l_4,m+1\,],[\,l_5,m+2\,],[\,l_6,m+2\,],[\,l_7,m+3\,],[\,l_8,m+3\,]\},

and let $\mathcal M_\mathbf C=\{W^{(v_j)}_{\,r_j-l_j+1,\;r_j}\mid [l_j,r_j]\in\mathbf C\}$ , where $[l_j,r_j]$ is the interval attached to the vertex $v_j$ ; that is, $\mathcal M_\mathbf C$ is the set of KR modules whose dominant monomials have the supports of $\mathbf C$ (for example, for type $A_n$ , $\mathcal M_{\mathbf C_{A_n}}=\{W^{(v_i)}_{\,m+\lfloor i/2\rfloor-l_i+2,\;m+\lfloor i/2\rfloor+1}\mid i=1,\dots,n\}$ ).

Conjecture 7.11 of Kanakubo-Koshevoy-Nakashima states: for every simply-laced type and every admissible choice of $m$ and $l_1<\cdots<l_n$ as above, the tensor product $\bigotimes_{W\in\mathcal M_\mathbf C}W$ is a simple $U_q(\widehat{\mathfrak g})$ -module. Prove or disprove this statement.

By the theorem of Hernandez, the statement is equivalent to: every pair of modules in $\mathcal M_\mathbf C$ strongly commutes (their pairwise tensor products are simple); in this form it is independent of the order of the factors, and a single admissible instance in which some pair fails to strongly commute disproves the conjecture. The required generality is that of the source: all five types; all admissible values of the parameters, with no bound on $n$ , $m$ or the gaps $l_{i+1}-l_i$ ; and the honest finite-dimensional KR modules of $U_q(\widehat{\mathfrak g})$ (not truncated q-character variants, unless the truncation is proved harmless). A complete answer may treat the five families separately.

References

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