Presentations of semigroup algebras of weighted trees

Manon, Christopher

doi:10.1007/s10801-009-0195-y

Cited by 12 publications

(8 citation statements)

References 9 publications

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“…In Section 5 we present a result in combinatorial commutative algebra that may be of independent interest: each phylogenetic tree specifies a degeneration of the Cox ring of X n to an algebra generated by Plücker monomials. This refines the results on sagbi bases in [17, §7], and it opens up the possibility of relating our Cox ring to the subalgebras studied by Howard et al [8] and Manon [10]. An important player in this connection should be the moduli space of rank two stable quasiparabolic bundles on P 1 with n points (cf.…”

Section: Introductionsupporting

confidence: 74%

Blow-ups of ${\bf P}^{n-3}$ at $n$ points and spinor varieties

Sturmfels¹,

Velasco²

2010

J. Commut. Algebra

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Work of Dolgachev and Castravet-Tevelev establishes a bijection between the 2 n−1 weights of the half-spin representations of so 2n and the generators of the Cox ring of the variety X n which is obtained by blowing up P n−3 at n points. We derive a geometric explanation for this bijection, by embedding Cox(X n ) into the even spinor variety (the homogeneous space of the even half-spin representation). The Cox ring of the blow-up X n is recovered geometrically by intersecting torus translates of the even spinor variety. These are higher-dimensional generalizations of results by Derenthal and Serganova-Skorobogatov on del Pezzo surfaces.

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Section: Introductionsupporting

confidence: 74%

Blow-ups of ${\bf P}^{n-3}$ at $n$ points and spinor varieties

Sturmfels¹,

Velasco²

2010

J. Commut. Algebra

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“…Remark 5.5. An identical construction to the one given above shows that the polytope P Γ ( r, L) studied in [Man10b] and [Man12] is a Newton-Okounkov bodies for R C, p ( r, L), where (C, p) is the stable curve of type Γ.…”

Section: ×V (γ)mentioning

confidence: 87%

“…Theorem 1.3 is utilized in [Man10b] and [Man12] to prove that the projective coordinate ring of the square of any effective line bundle on the moduli stack M C, p (SL 2 (C)) is generated by its degree 1 elements, and is a Koszul algebra for generic C, p. This result follows from the analysis of P Γ ( r, L) for particular wellchosen trivalent graphs. We follow a similar strategy here, by studying a particular polytope P Γg,n , we prove the following.…”

Section: Introductionmentioning

confidence: 99%

The algebra of conformal blocks

Manon

2018

J. Eur. Math. Soc.

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For each simply connected, simple complex group G we show that the direct sum of all vector bundles of conformal blocks on the moduli stack Mg,n of stable marked curves carries the structure of a flat sheaf of commutative algebras. The fiber of this sheaf over a smooth marked curve (C, p) agrees with the Cox ring of the moduli of quasi-parabolic principal G−bundles on (C, p). We use the factorization rules on conformal blocks to produce flat degenerations of these algebras. In the SL 2 (C) case, these degenerations result in toric varieties which appear in the theory of phylogenetic statistical varieties, and the study of integrable systems in the moduli of rank 2 vector bundles. We conclude with a combinatorial proof that the Cox ring of the moduli stack of quasi-parabolic SL 2 (C) principal bundles over a generic curve is generated by conformal blocks of levels 1 and 2 with relations generated in degrees 2, 3, and 4.

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“…Proof. By a general theorem due to Zhelobenko [Zhe73] (see also [Man10]), the space [V (λ) ⊗ V (ω i ) ⊗ V (µ)] SLm(C) can be identified with a subspace of a weight space of V (ω i ) = i (C m ). As we remarked above, all weight spaces of these representations are multiplicity free.…”

Section: Conformal Blocks and Invariants In Tensor Productsmentioning

confidence: 99%

Conformal blocks, Berenstein–Zelevinsky triangles, and group-based models

Kubjas

Manon

2014

J Algebr Comb

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Abstract. Work of Buczyńska, Wiśniewski, Sturmfels and Xu, and the second author has linked the group-based phylogenetic statistical model associated with the group Z/2Z with the Wess-Zumino-Witten (WZW) model of conformal field theory associated to SL 2 (C). In this article we explain how this connection can be generalized to establish a relationship between the phylogenetic statistical model for the cyclic group Z/mZ and the WZW model for the special linear group SLm(C). We use this relationship to also show how a combinatorial device from representation theory, the Berenstein-Zelevinsky (BZ) triangles, correspond to elements in the affine semigroup algebra of the Z/3Z phylogenetic statistical model.

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Presentations of semigroup algebras of weighted trees

Cited by 12 publications

References 9 publications

Blow-ups of ${\bf P}^{n-3}$ at $n$ points and spinor varieties

Blow-ups of ${\bf P}^{n-3}$ at $n$ points and spinor varieties

The algebra of conformal blocks

Conformal blocks, Berenstein–Zelevinsky triangles, and group-based models

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