88 pages, AMS-LaTeX, uses xypic 3.7 package; v2: minor typos fixed, definitions of D_P(V) in 10.2 and of n_P(V) in 24.1 corrected and their usage revised; v3: minor typos fixed, proof of Prop. 16.2 simplified, minor improvements in exposition; v4: many typos fixed and improvements in exposition
L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of a locally symmetric space. We define the micro-support of an L-module; it is a set of irreducible modules for the Levi quotients of the parabolic $\mathbb Q$-subgroups associated to the strata. We prove a vanishing theorem for the global cohomology of an L-module in term of the micro-support. We calculate the micro-support of the middle weight profile weighted cohomology and the middle perversity intersection cohomology L-modules. (For intersection cohomology we must assume the $\mathbb Q$-root system has no component of type D, E, or F.) Finally we prove a functoriality theorem concerning the behavior of micro-support upon restriction of an L-module to the pre-image of a Satake stratum. As an application we settle a conjecture made independently by Rapoport and by Goresky and MacPherson, namely, that the intersection cohomology (for either middle perversity) of the reductive Borel-Serre compactification of a Hermitian locally symmetric space is isomorphic to the intersection cohomology of the Baily-Borel-Satake compactification. We also obtain a new proof of the main result of Goresky, Harder, and MacPherson on weighted cohomology as well as generalizations of both of these results to general Satake compactifications with equal-rank real boundary components.
An overview of the theory of L-modules and the above conjecture, as well as an application to the cohomology of arithmetic groups, can be found in math.RT/0112250 .
Let $A$ be the one point extension of an algebra $B$ by a projective $B$-module. We prove that the extension of a given support $τ$-tilting $B$-module is a support $τ$-tilting $A$-module; and, conversely, the restriction of a given support $τ$-tilting $A$-module is a support $τ$-tilting $B$-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support $τ$-tilting modules.
In this article, we prove that induced modules of support $τ$-tilting modules over blocks of finite groups satisfying inertial-invariant condition are also support $τ$-tilting modules.
Let $B$ be the one-point extension algebra of $A$ by an $A$-module $X$. We proved that every support $τ$-tilting $A$-module can be extended to be a support $τ$-tilting $B$-module by two different ways. As a consequence, it is shown that there is an inequality $$|\stilt B|\geqslant 2|\stilt A|.$$
Support $τ$-tilting modules correspond to some classes of categorical objects bijectively, such as two-term tilting complexes for any finite dimensional symmetric algebra. This fact motivates us to classify support $τ$-tilting modules over blocks of finite groups. Therefore we classify support $τ$-tilting modules over particular blocks of finite groups by using the modular representation theoretical approaches including Clifford theory, Green's indecomposability theorem and so on.
C. Ingalls and H. Thomas defined support tilting modules for path algebras. From tau-tilting theory introduced by T. Adachi, O. Iyama and I. Reiten, a partial order on the set of basic tilting modules defined by D. Happel and L. Unger is extended as a partial order on the set of support tilting modules. In this paper, we study a combinatorial relationship between the poset of basic tilting modules and basic support tilting modules. We will show that the subposet of tilting modules is uniquely determined by the poset structure of the set of support tilting modules.
In this paper we study the poset of basic tilting $kQ$-modules when $Q$ is a Dynkin quiver, and the poset of basic support $τ$-tilting $kQ$-modules when $Q$ is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if $Q$ is of types $\mathbb{A}_{1}, \mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a nonlinear orientation and the second poset is a distributive lattice if and only if $Q$ is of type $\mathbb{A}_{1}$.
In this article we study support $τ$-tilting modules, semibricks and more over blocks of group algebras. Let $k$ be an algebraically closed field of characteristic $p>0$, $\tilde{G}$ a finite group and $G$ a normal subgroup of $\tilde{G}$. Moreover, let $\tilde{B}$ be a block of $k\tilde{G}$ and $B$ a block of $kG$ covered by $\tilde{B}$. We show that, under certain conditions for the factor group $\tilde{G}/G$ and $B$, induced modules and extending modules of support $τ$-tilting modules and semibricks over $B$ are also the ones over $\tilde{B}$, respectively.
Let $\tilde{G}$ be a finite group and $G$ a normal subgroup of $\tilde{G}$. In this paper, we give a necessary and sufficient condition for $\mathrm{Ind}_G^{\tilde{G}}M$ to be a support $τ$-tilting $k\tilde{G}$-module for a $kG$-module $M$. Moreover, we give the block version of the result.
We introduce the notion of essential support of a simple Gelfand-Tsetlin $\mathfrak{gl}_n$-module as an important tool towards understanding the character formula of such module. This support detects the weights in the module having maximal possible Gelfand-Tsetlin multiplicities. Using combinatorial tools we describe the essential supports of the simple socles of the universal tableaux modules. We also prove that every simple Verma module appears as a socle of a universal tableaux module and hence obtain a description of the essential supports of all simple Verma modules. As a consequence, we…
Let $(\mbox{mod} Λ',\mbox{mod} Λ,\mbox{mod} Λ'')$ be a recollement of abelian categories for artin algebras $Λ'$, $Λ$ and $Λ''$. Under certain conditions, we present an explicit construction of gluing of (support) $τ$-tilting modules in $\mbox{mod} Λ$ with respect to (support) $τ$-tilting modules in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ respectively; conversely, we study the construction of (support) $τ$-tilting modules in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ obtained from (support) $τ$-tilting modules in $\mbox{mod} Λ$.
For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Geiß~ et al. (2016) introduced a generalized preprojective algebra $Π(C, D)$. We study tilting modules and support $τ$-tilting modules for the generalized preprojective algebra $Π(C, D)$ and show that there is a bijection between the set of all cofinite tilting ideals of $Π(C,D)$ and the corresponding Weyl group $W(C)$ provided that $C$ has no component of Dynkin type. When $C$ is of Dynkin type, we also establish a bijection between the set of all basic support $τ$-tilting $Π(C,D)$-modules and the corresponding Weyl group $…
In this paper, as the first step towards classification of simple weight modules with finite dimensional weight spaces over Witt algebras $W_n$, we explicitly describe supports of such modules. We also obtain some descriptions on the support of an arbitrary simple weight module over a $\Z^n$-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\oplus_{α\in\Z^n}\mathfrak {g}_α$ with respect to the abelian subalgebra $\mathfrak {g}_0$, with the property $[\mathfrak{g}_α,\mathfrak {g}_β]= \mathfrak {g}_{α+β}$ for all $α,β\in\Z^n$, $α\neq β$ (this class contains the algebra $W_n$).
A recent work by Lesnick and Wright proposed a visualisation of $2$D persistence modules by using their restrictions onto lines, giving a family of $1$D persistence modules. We give a constructive proof that any $1$D persistence module with finite support can be found as a restriction of some indecomposable $2$D persistence module with finite support. As consequences of our construction, we are able to exhibit indecomposable $2$D persistence modules whose support has holes as well as an indecomposable $2$D persistence module containing all $1$D persistence modules with finite support as line r…
Motivated by $τ$-tilting theory developed by Adachi, Iyama and Reiten, for a finite-dimensional algebra $Λ$ with action by a finite group $G$, we introduce the notion of $G$-stable support $τ$-tilting modules. Then we establish bijections among $G$-stable support $τ$-tilting modules over $Λ$, $G$-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective $Λ$-modules, and $G$-stable functorially finite torsion classes in the category of finitely generated left $Λ$-modules. In the case when $Λ$ is the endomorphism of a $G$-stable cluster-tilt…
This paper fixes a gap in my article "Support varieties of non-restricted modules over Lie algebras of reductive groups" pointed out to me by J.C. Jantzen. It was written several years ago, but never widely circulated.
Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of the nilpotent cone of $G$, defined in terms of cohomology for the first Frobenius kernel $G_1$. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for $G = \mathrm{SL}_n$ in earlier work of the second author. In this paper, we show that for any $G$, the support variety of a til…
Let $\tilde{G}$ be a finite group, $G$ a normal subgroup of $\tilde{G}$ and $k$ an algebraically closed field of characteristic $p>0$. The first main result in this paper is to show that support $τ$-tilting $k\tilde{G}$-modules satisfying some properties are support $τ$-tilting modules as $kG$-modules too. As the second main result, we give equivalent conditions for support $τ$-tilting $k\tilde{G}$-modules to satisfy the above properties, and show that the set of the support $τ$-tilting $k\tilde{G}$-modules with the properties is isomorphic to the set of $\tilde{G}$-invariant support $τ$-tilti…
We consider the support varieties of Demazure modules, certain $B$-modules important in the representation theory of reductive groups. In many cases we are able to compute these support varieties over $B_1$, the first Frobenius kernel of a Borel subgroup, and relate them to orbital varieties. We provide a complete calculation of the support varieties when the underlying algebraic group has type $A_1$ or $A_2$. We also consider the saturation ($G$ orbit of) $B_1$ support varieties for Demazure modules and show that the inclusion relations among these varieties are compatible with the Bruhat ord…
We study (support) $τ$-tilting modules over the trivial extensions of finite dimensional algebras. More precisely, we construct two classes of (support)$τ$-tilting modules in terms of the adjoint functors which extend and generalize the results on (support) $τ$-tilting modules over triangular matrix rings given by Gao-Huang.
The goal of this paper is to compute the supports of simple modules in the categories $\mathcal{O}$ for the rational Cherednik algebras associated to groups $G(\ell,1,n)$. For this we compute some combinatorial maps on the set of simples: wall-crossing bijections and a certain $\mathfrak{sl}_\infty$-crystal associated to a Heisenberg algebra action on a Fock space.
We give a necessary and sufficient condition for that the support $τ$-tilting poset of a finite dimensional algebra is isomorphic to the poset of symmetric group with weak order. Moreover we show that there are infinitely many finite dimensional algebras whose support $τ$-tilting posets are isomorphic to the poset of symmetric group with weak order.
Let $kG$ be the group algebra of a finite group scheme defined over a field $k$ of characteristic $p>0$. Associated to any closed subset $V$ of the projectivized prime ideal spectrum $\operatorname{Proj} \operatorname{H}^*(G,k)$ is a thick tensor ideal subcategory of the stable category of finitely generated $kG$-module, whose closure under arbitrary direct sums is a localizing tensor ideal in the stable category of all $kG$-modules. The colocalizing functor from the big stable category to this localizing subcategory is given by tensoring with an idempotent module $\mathcal{E}$. A property of …
The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper exhibits the number of support-tilting modules for any Dynkin algebra. Since the support-tilting modules for a Dynkin algebra correspond bijectively to the non-crossing partitions of the same type, the calculations presented here may also be considered as a categorification of results concerning non-crossing partitions.
We introduce the notions of Gorenstein projective $τ$-rigid modules, Gorenstein projective support $τ$-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi-Iyama-Reiten's bijection theorem on support $τ$-tilting modules. More precisely, for an algebra $Λ$, We prove that there is a bijection between the set of Gorenstein projective support $τ$-tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-$τ$-tilting finite algebras and show that $Λ$ is CM-$τ$-tilting finite if and only if…
An irreducible weight module of an affine Kac-Moody algebra $\mathfrak{g}$ is called dense if its support is equal to a coset in $\mathfrak{h}^{*}/Q$. Following a conjecture of V. Futorny about affine Kac-Moody algebras $\mathfrak{g}$, an irreducible weight $\mathfrak{g}$-module is dense if and only if it is cuspidal (i.e. not a quotient of an induced module). The conjecture is confirmed for $\mathfrak{g}=A_{2}^{\left(1\right)}$, $A_{3}^{\left(1\right)}$ and$A_{4}^{\left(1\right)}$ and a classification of the supports of the irreducible weight $\mathfrak{g}$-modules obtained. For all $A_{n}^{\…
The class of support $τ$-tilting modules was introduced recently by Adachi, Iyama and Reiten. These modules complete the class of tilting modules from the point of view of mutations. Given a finite dimensional algebra $A$, we study all basic support $τ$-tilting $A$-modules which have given basic $τ$-rigid $A$-module as a direct summand. We show that there exist an algebra $C$ such that there exists an order-preserving bijection between these modules and all basic support $τ$-tilting $C$-modules; we call this process $τ$-tilting reduction. An important step in this process is the formation of $…
In this paper we give necessary and sufficient conditions for the variety of a simple module over a (D,A)-stacked monomial algebra to be nontrivial. This class of algebras was introduced in [Green and Snashall, The Hochschild cohomology ring modulo nilpotence of a stacked monomial algebra, Colloq. Math. 105 (2006), 233-258] and generalizes Koszul and D-Koszul monomial algebras. As a consequence we show that if the variety of every simple module over such an algebra is nontrivial then the algebra is D-Koszul. We give examples of (D,A)-stacked monomial algebras which are not selfinjective but ne…
Let $G$ be a reductive algebraic group scheme defined over $\mathbb{F}_p$ and let $G_1$ denote the Frobenius kernel of $G$. To each finite-dimensional $G$-module $M$, one can define the support variety $V_{G_1}(M)$, which can be regarded as a $G$-stable closed subvariety of the nilpotent cone. A $G$-module is called a tilting module if it has both good and Weyl filtrations. In 1997, it was conjectured by J.E. Humphreys that when $p\geq h$, the support varieties of the indecomposable tilting modules coincide with the nilpotent orbits given by the Lusztig bijection. In this paper, we shall verif…
Let $\mathcal{D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object $R$. Let $Λ=\operatorname{End}_{\mathcal{D}}R$ be the endomorphism algebra of $R$. We introduce the notion of mutation of maximal rigid objects in the two-term subcategory $R\ast R[1]$ via exchange triangles, which is shown to be compatible with mutation of support $τ$-tilting $Λ$-modules. In the case that $\mathcal{D}$ is the cluster category arising from a punctured marked surface, it is shown that the graph of mutations of support $τ$-tilting $Λ$-modules is isomorphic to the graph of fli…
The representations of some Hopf algebras have curious behavior: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. We explain a family of examples of such Hopf algebras and their modules, and classify left, right, and two-sided ideals in their stable module categories.
The proof of Theorem 5.12 in [C.M. Drupieski, On injective modules and support varieties for the small quantum group, Int. Math. Res. Not. 2011 (2011), 2263-2294] does not make sense as written because the algebra $u_ζ(\mathfrak{b}_α^+)$ need not be a Hopf subalgebra of $u_ζ(\mathfrak{b}^+)$ unless $α$ is a simple root. This note describes how the proof should be modified to work around this fact.
We consider a finite dimensional strongly $G$-graded algebra $A$ with { self-injective} $1$-component $B$, and in our main result we prove that the induction from $B$ to $A$ of a basic support $τ$-tilting pair of $B$-modules is a support $τ$-tilting pair $(M,P)$ of $A$-modules if and only if $M$ is $G$-invariant. A similar statement holds for the restriction from $A$ to $B$, so our results may be viewed as Clifford and Maschke type theorems for $2$-term silting complexes. We also give applications to semibricks and the associated wide subcategories.
Let $\frak g$ be a reductive Lie algebra over an algebraically closed field of characteristic 0 and $\frak k$ be a reductive in $\frak g$-subalgebra. Let $M$ be a finitely generated (possibly, infinite-dimensional) $\frak g$-module. We say that $M$ is a $(\frak g, \frak k)$-module if $M$ is a direct sum of a (possibly, infinite) amount of simple finite-dimensional $\frak k$-modules. We say that $M$ is of finite type if $M$ is a $(\frak g, \frak k)$-module and Hom$_\frak k(V, M)<\infty$ for any simple $\frak k$-module $V$. Let $X$ be a variety of all Borel subalgebras of $\frak g$. Let $M$ be a…
We investigate cohomological support varieties for finite-dimensional Lie superalgebras defined over fields of odd characteristic. Verifying a conjecture from our previous work, we show the support variety of a finite-dimensional supermodule can be realized as an explicit subset of the odd nullcone of the underlying Lie superalgebra. We also show the support variety of a finite-dimensional supermodule is zero if and only if the supermodule is of finite projective dimension. As a consequence, we obtain a positive characteristic version of a theorem of Bøgvad, showing that if a finite-dimensiona…
A Gelfand-Tsetlin tableau $T(v)$ induces a character $χ_v$ of the Gelfand-Tsetlin subalgebra $Γ$ of $U = U(\mathfrak{gl}(n,\mathbb C))$. By a theorem due to Ovsienko, for each tableau $T(v)$ there exists a finite number of nonisomorphic irreducible Gelfand-Tsetlin modules with $χ_v$ in its support, though explicit examples of such modules are only known for special families of characters. In this article we build a family of Gelfand-Tsetlin modules parametrized by characters, such that each character appears in its corresponding module. We also find the support of these modules, with multiplic…
Support varieties for any finite dimensional algebra over a field were introduced by Snashall-Solberg using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected…
In this paper, we deal with the $\mathcal{U}(\mathfrak{g})$-action on a $\mathfrak{g}$-module on which a larger algebra $\mathcal{A}$ acts irreducibly. Under a mild condition, we will show that the support of the $\mathcal{Z}(\mathfrak{g})$-action is a union of affine subspaces in the dual of a Cartan subalgebra modulo the Weyl group action. As a consequence, we propose a definition of a Cartan subalgebra for such a $\mathfrak{g}$-module. The support of the $\mathcal{Z}(\mathfrak{g})$-module is an algebraic counterpart of the support of the measure in the irreducible decomposition of a unitary…
Let $Γ$ be a split extension of a finite-dimensional algebra $Λ$ by a nilpotent bimodule $_ΛE_Λ$, and let $(T,P)$ be a pair in $\modΛ$ with $P$ projective. We prove that $(T\otimes_ΛΓ_Γ, P\otimes_ΛΓ_Γ)$ is a support $τ$-tilting pair in $\mod Γ$ if and only if $(T,P)$ is a support $τ$-tilting pair in $\mod Λ$ and $\Hom_Λ(T\otimes_ΛE,τT_Λ)=0=\Hom_Λ(P,T\otimes_ΛE)$. As applications, we obtain a necessary and sufficient condition such that $(T\otimes_ΛΓ_Γ, P\otimes_ΛΓ_Γ)$ is support $τ$-tilting pair for a cluster-tilted algebra $Γ$ corresponding to a tilted algebra $Λ$; and we also get that if $T_…
We study support $τ$-tilting modules over preprojective algebras of Dynkin type. We classify basic support $τ$-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and other important objects in representation theory. We also study $g$-matrices of support $τ$-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of $g$-matrices and prove that cones given by $g$-matrices co…
Let $u_ζ(g)$ denote the small quantum group associated to the simple complex Lie algebra $g$, with parameter $q$ specialized to a primitive $\ell$-th root of unity $ζ$ in the field $k$. Generalizing a result of Cline, Parshall and Scott, we show that if $M$ is a finite-dimensional $u_ζ(g)$-module admitting a compatible torus action, then the injectivity of $M$ as a module for $u_ζ(g)$ can be detected by the restriction of $M$ to certain root subalgebras of $u_ζ(g)$. If the characteristic of $k$ is positive, then this injectivity criterion also holds for the higher Frobenius--Lusztig kernels $U…
We define relative support varieties with respect to some fixed module over a finite dimensional algebra. These varieties share many of the standard properties of classical support varieties. Moreover, when introducing finite generation conditions on cohomology, we show that relative support varieties contain homological information on the modules involved. As an application, we provide a new criterion for a selfinjective algebra to be of wild representation type.
In a previous paper we constructed rank and support variety theories for "quantum elementary abelian groups," that is, tensor products of copies of Taft algebras. In this paper we use both variety theories to classify the thick tensor ideals in the stable module category, and to prove a tensor product property for the support varieties.
In continuation of work begun in \cite{FR}, we study in this article those Auslander--Reiten components of the algebras $\Dist(G_r)$ that contain simple modules or baby Verma modules, where $\Dist(G_r)$ is the algebra of distributions of the $r$-th Frobenius kernel of the smooth reductive group $G$, defined over an algebraically closed field of positive characteristic.
We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by a cocycle. We develop a theory of support varieties for these Hopf algebras to use as a tool in our investigations.
Mutation of τ-tilting modules is a basic operation to construct a new support τ-tilting module from a given one by replacing a direct summand. The aim of this paper is to give a positive answer to the question posed in [AIR, Question 2.31] about mutation of τ-tilting modules.
Happel and Unger reconstructed hereditary algebras from their posets of tilting modules. Inspired by this result, we try removing the assumption to be hereditary. However, it would be unfortunately fail in general: e.g. every selfinjective algebra has the poset consisting of only one point. Therefore, we should consider a generalization of the Happel-Unger's result for posets of support $τ$-tilting modules, which contains those of tilting modules. In this paper, we spotlight finite dimensional algebras whose support $τ$-tilting posets coincide with those of tree quiver algebras and give a full…
The theory of $τ$-tilting was introduced by Adachi--Iyama--Reiten; one of the main results is a bijection between support $τ$-tilting modules and torsion classes. We are able to generalise this result in the context of the higher Auslander--Reiten theory of Iyama. For a finite-dimensional algebra $A$ with 2-cluster-tilting subcategory $\mathcal{C}\subseteq\mathrm{mod}A$, we are able to find a correspondence between support $τ_2$-tilting $A$-modules and torsion pairs in $\mathcal{C}$ satisfying an additional functorial finiteness condition.
We consider rational representations of a connected linear algebraic group $\mathbb G$ over a field $k$ of positive characteristic $p > 0$. We introduce a natural extension $M \mapsto Π(\mathbb G)_M$ to $\mathbb G$-modules of the $π$-point support theory for modules $M$ for a finite group scheme $G$ and show that this theory is essentially equivalent to the more "intrinsic" and "explicit" theory $M \mapsto \mathbb P\mathfrak C(\mathbb G)_M$ of supports for an algebraic group of exponential type, a theory which uses 1-parameter subgroups $\mathbb G_a \to \mathbb G$. We extend our support theory…
We construct two families of refinements of the (projectivized) support variety of a finite dimensional module $M$ for a finite group scheme $G$. For an arbitrary finite group scheme, we associate a family of {\it non maximal rank varieties} $Γ^j(G)_M$, $1\leq j \leq p-1$, to a $kG$-module $M$. For $G$ infinitesimal, we construct a finer family of locally closed subvarieties $V^{\ul a}(G)_M$ of the variety of one parameter subgroups of $G$ for any partition $\ul a$ of $\dim M$. For an arbitrary finite group scheme $G$, a $kG$-module $M$ of constant rank, and a cohomology class $ζ$ in $\HHH^1(G…
Let $λ$, $μ$, $ν$ and $ρ$ be dominant weights of $\mathfrak{sl_n}$ satisfying $λ+ μ= ν+ ρ$. Let $V_λ$ denote the highest weight module corresponding to $λ$. Lam, Postnikov, Pylyavskyy conjectured a sufficient condition for $V_λ \otimes V_μ$ to be contained in $V_ν \otimes V_ρ$ as $\mathfrak{sl_n}$-modules. In this note we prove a weaker version of the conjecture. Namely we prove that under the conjectured conditions every irreducible $\mathfrak{sl_n}$-module which appears in the decomposition of $V_λ \otimes V_μ$ does appear in the decomposition of $V_ν \otimes V_ρ$.
