Presentations of semigroup algebras of weighted trees

Christopher Manon 0 0 The author was supported by the NSF FRG grant DMS-0554254. The author will include these results in his doctoral thesis. C. Manon ( ) Department of Mathematics, University of Maryland, College Park , MD 20742, USA We study presentations for subalgebras of invariants of the coordinate algebras of binary symmetric models of phylogenetic trees studied by Buczynska and Wisniewski in (J. Eur. Math. Soc. 9:609-635, 2007). These algebras arise as toric degenerations of projective coordinate rings of the moduli of weighted points on the projective line, and projective coordinate rings of the moduli of quasiparabolic semisimple rank two bundles on the projective line. Let T be an abstract trivalent tree with leaves V (T ), edges E(T ), and non-leaf vertices I (T ), by trivalent we mean that the valence of v I (T ) is always three. Let ei be the unique edge incident to the leaf i V (T ). Let Y be the unique trivalent tree with three leaves. For each v I (T ) we pick an injective map iv : Y T , sending the unique member of I (Y ) to v. We denote the members of E(Y ) by E, F , and G. We say that two leaves e, f V (T ) are paired if they are connected to a common internal vertex. Members of V (T ) which are not paired are called lone leaves. We will be concerned with properties of weightings of trivalent trees, defined as a functions - Z0. The maps iv define pull-back operations on weightings by the formulas Definition 1.1 Let ST be the graded semigroup where ST [k] is the set of weightings which satisfy the following conditions. These are referred to as the triangle inequalities. (2) iv()(E) + iv()(F ) + iv()(G) is even. (3) Note that because the triangle inequalities hold for the integers iv()(E), iv()(F ), and iv()(G) if and only if a triangle exists with these side lengths, the condition is symmetric in E, F , and G. In [8] Speyer and Sturmfels show that the semigroup algebras C S deformations of Gr2( C[nT) ]wmhearyebne =rea|Vliz(eTd)a|,s fporrotjhecetiPvlecckoeorrdeimnabteedrdiinnggs. oTfhflisatsteomriicgroup is also multigraded, with the grading given by the weights (ei ) on the leaf edges of the tree. For a vector of non-negative integers r = (r1, . . . , rn) we let ST [r] be the set of weightings ST with (ei ) = ri . Definition 1.2 Let r : V (T ) Z0 be a vector of non-negative integers. Let ST (r) be the multigraded subsemigroup of ST formed by the pieces ST [kr]. In general we will focus on weightings of trivalent trees such that the vector of edge weights r has an even total sum, because of the following proposition. The proof provides a nice introduction to the study of weights on trivalent trees. Proposition 1.3 Let T be a trivalent tree. If r has an odd total sum, then there is no weighting of T satisfying the parity condition with edges weighted by r. Proof Note that this is true by definition for T = Y. Suppose that the result holds for every trivalent tree with n 1 leaves, and consider a T with n leaves. Pick paired leaves e, f in V (T ), and let T be the trivalent tree obtained by forgetting e and f , and the edges connected to them. Let g be the internal edge of T which shares a vertex with f and g. Note that we may consider g a leaf of T . Any weighting which satisfies parity also defines a weighting of T by restriction. By the induction hypothesis, |T weights an even number of V (T ) with odd numbers. There are two cases, if g is weighted odd then by parity only one of e or f may be weighted odd. If g is weighted even, then either both e and f are weighted odd, or neither is weighted It follows from work in [8] that graded algebras C[ST (r)] are homogeneous coordinate rings for projective embeddings of flat toric deformations of Gr2(Cn)//rT , the weight variety of the Grassmannian of 2-planes associated to r, or equivalently Mr, the moduli space of r-weighted points on CP 1 (see [6] for this connection). In [6] this degeneration is used to construct presentations of the projective coordinate ring of Mr for the Plcker embedding, and it is shown that these algebras are generated in degree 1 and have relations generated by quadrics and cubics for certain T and r. This is the starting point for the present paper. Forgetting the grading for a moment, geometrically ST is the semigroup of lattice points in a cone PT in R|E(T )|. The inequalities defining PT are given by the triangle inequalities, and the parity condition defines a certain sublattice of Z|E(T )|. Let T1 and T2 be trivalent trees with N1 and N2 leaves, respectively. Identify the leaf 1 from T2 with the leaf N1 from T1, relabelling the leaves of T2 as follows, 2 N1 + 1, . . . , N2 N1 + N2 1. This creates a tree with a unique vertex of valence 2, replace this vertex and both of its incident edges with a single edge, the resulting tree T1 T2 is trivalent. We call this operation merging, see Figure 1 for an example. Let i V (T ), and denote the projection onto the ei -th component of R|E(T )| by i . It is simple to check that Where the right hand side is the fibered product of the polytopes PT1 and PT2 over the maps N1 and 1. In particular this implies that all PT are fibered products of copies of PY . This is reminiscent of the theory of moduli of structures on orientable surfaces, where structures on a surface of high genus can be glued together from structures on three-punctured spheres over a pair-of-pants decomposition. The reason for this resemblance is not entirely accidental, see [5] for a moduli-of-surfaces interpretation of spaces associated to the semigroup ST . Buczynska and Wisniewski define merging in [3], where they show that a similar fibered product formula holds for a class of semigroups of weightings which we will now introduce. Definition 1.4 For a trivalent tree T let ( T ) be the polytope in R|E(T )| formed by the convex hull of weightings such that (e) {0, 1} for all e E(T ), and iv()(E) + iv()(F ) + iv()(G) 2Z for all v I (T ). It is shown in [3] (proposition 1.13) that ( T ) is a fibered product of |I (T )| copies of (Y ). The lattice point semigroup of L ( T ) = ( T ) + . . . + ( T ) is isomorphic to the following semigroup. Definition 1.5 Let L be a positive integer. Let SL be the graded semigroup where T SL [k] is the set of weightings of T which satisfy T (1) For all v I (T ) the numbers iv()(E), iv()(F ) and iv()(G) satisfy the triangle inequalities. (2) iv()(E) + iv()(F ) + iv()(G) is even. This last condition is referred to as the level condition. Note that S1 has a fibered product decomposition into copies SY1 in a way completely analogoTus to ST . To see that the lattice points of ( T ) correspond with the first graded piece of S1 , one need only use the fibered product decomposition of both T objects. We observe that the lattice points of (Y ) are given by the degree 1 members of SY1 . In [3] Buczynska and Wisniewski study the algebras C S1 ], proving that they are all deformation equivalent. However, they do no (...truncated)