Abstract
We study the closure of the graph of the birational map from a projective space to a Grassmannian. We provide explicit description of the graph closure and compute the fibers of the natural projection to the Grassmannian. We construct embeddings of the graph closure to the projectivizations of certain cyclic representations of a degenerate special linear Lie algebra and study algebraic and combinatorial properties of these representations. In particular, we describe monomial bases, generalizing the FFLV bases. The proof relies on combinatorial properties of a new family of poset polytopes, which are of independent interest. As a consequence we obtain flat toric degenerations of the graph closure studied by Borovik, Sturmfels and Sverrisdóttir.
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Acknowledgements
We are grateful to Alexander Kuznetsov and Igor Makhlin for useful discussions and correspondence.
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Open access funding provided by Tel Aviv University.
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The main body of the paper is written by E.F. The appendix is written by E.F. and W.S.
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Appendix A: A Proof of the Minkowski Property (in Collaboration with Wojciech Samotij)
Appendix A: A Proof of the Minkowski Property (in Collaboration with Wojciech Samotij)
Let P be a finite poset. In Sect. 1, we defined the sets \(S(m,M)\subseteq {\mathbb Z}_{\ge 0}^P\).
Theorem A.1
\(S(m,M+1) = S(m,M) + S(0,1)\) for all integers \(m, M \ge 0\).
Proof
The inclusion \(S(m,M+1) \supseteq S(m,M) + S(0,1)\) is immediate. Indeed, if \(x \in S(m,M)\) and \(y \in S(0,1)\), then, for all \(P' \subseteq P\),
and thus \(x+y = (x_\alpha + y_\alpha )_{\alpha \in P} \in S(m,M+1)\). We will prove that the reverse inclusion is a consequence of duality of linear programming [32].
Given a vector \(z \in \mathbb {Z}^P\), define
and note that a vector \(z \in \mathbb {Z}_{\ge 0}^P\) belongs to S(m, M) if and only if \(M(z) \le M\). We will prove the following assertion. We write \(e_\delta \) to denote the vector \((\mathbbm {1}_{\beta = \delta })_{\beta \in P} \in \mathbb {Z}_{\ge 0}^P\) (i.e. the \(\delta \)-th coordinate of \(e_\delta \) is equal to one and all other coordinates vanish). \(\square \)
Lemma A.2
For every \(z \in \mathbb {Z}^P\) such that \(M(z) > 0\), there exists some \(\delta \in P\) such that \(M(z-e_\delta ) \le M(z) - 1\).
We first show the lemma yields the inclusion \(S(m,M+1) \subseteq S(m,M) + S(0,1)\). Fix an arbitrary \(z \in S(m,M+1)\). We may clearly assume that \(z \notin S(m,M)\), as otherwise there is nothing to prove. Additionally, we may assume that \(z_\alpha \ge 1\) for all \(\alpha \in P\), as otherwise we can replace P with its subposet induced by the set \(\{\alpha \in P: z_\alpha \ge 1\}\) and remove from z all its zero coordinates. Indeed, this operation does not change the value of M(z). Since \(z \notin S(m,M)\), we have \(M(z) = M+1 > 0\) and thus we may invoke Lemma A.2 to find a \(\delta \in P\) satisfying the assertion of the lemma. Clearly, \(e_\delta \) belongs to S(0, 1) and \(z - e_\delta \in \mathbb {Z}_{\ge 0}^P\) thanks to our assumption that \(z \ge 1\). Further, \(z - e_\delta \in S(m,M)\), as \(M(z - e_\delta ) \le M(z) - 1 = M\). We now turn to the proof of Lemma A.2.
Let \(\bar{P}\) be the poset obtained from P by adding to it two new elements \(\sigma \) and \(\tau \) so that \(\sigma \le \alpha \le \tau \) for all \(\alpha \in P\). Fix an arbitrary vector \(z \in \mathbb {Z}^P\) and consider the following integer program (P):
Claim A.2.1
The value of (P) is M(z)
Proof
Let \(P'\) be an arbitrary subposet of P and let \(\mathcal {C}_{P'}\) be a collection of \(w(P')\) pairwise-disjoint chains whose union is \(P'\). Adding \(\sigma \) and \(\tau \) to each chain in \(\mathcal {C}_{P'}\) yields a collection of \(w(P')\) chains of the form \(\sigma< \alpha _1< \cdots< \alpha _\ell < \tau \), where \(\alpha _1, \dotsc , \alpha _\ell \in P'\). It is not hard to verify that the vector \((f_{\alpha \beta })_{\alpha < \beta }\) such that \(f_{\sigma \alpha _1} = \cdots = f_{\alpha _\ell \tau } = 1\) for every chain in \(\mathcal {C}_{P'}\) and \(f_{\alpha \beta } = 0\) otherwise is a feasible solution to (P) whose cost is
We conclude that the value of (P) is at least the maximum of Eq. A.1 over all \(P' \subseteq P\), that is, M(z).
Conversely, observe that every feasible solution \(f = (f_{\alpha \beta })_{\alpha < \beta }\) to (P) corresponds to a collection \(\mathcal {C}_f\) of \(|\{\alpha \in P: f_{\alpha \tau } = 1\}|\) chains of the form \(\sigma< \alpha _1< \cdots< \alpha _\ell < \tau \), where \(\alpha _1, \dotsc , \alpha _\ell \in P\) and \(f_{\sigma \alpha _1} = \cdots = f_{\alpha _\ell \tau } = 1\), such that every \(\alpha \in P\) belongs to at most one chain in \(\mathcal {C}_f\). Let \(P_f' \subseteq P\) be the set of elements of P that appear on some chain in \(\mathcal {C}_f\) and note that the value of the cost function at f is
where the first inequality holds as \(P_f'\) is a union of \(|\{ \alpha \in P: f_{\alpha \tau } = 1\}|\) chains. \(\square \)
Let us now write (P) in a more compact (but equivalent) form. Set \(R:= \{(\alpha ,\gamma ) \in \bar{P}^2: \alpha < \gamma \}\) and define the following two \(|P| \times |R|\) matrices:
Writing \(z_\tau := -m\), we may reformulate (P) as follows:
(Note that the constraint \(f \le 1\) in the original formulation of (P) follows from the constraints \(A'f \le 1\), \(Af = 0\), and \(f \ge 0\).)
Claim A.2.2
The \(2|P| \times |R|\) matrix \(\begin{pmatrix}A \\ A'\end{pmatrix}\) is totally unimodular.
Proof
Let B be an arbitrary square submatrix of \(\begin{pmatrix}A \\ A'\end{pmatrix}\) and let \(Q \subseteq P\) and \(Q' \subseteq P\) be the rows of A and \(A'\), respectively, that appear in B. Let \(B'\) be the matrix obtained from B by the following row operations: For every \(\beta \in Q \cap Q'\), subtract \(A'_\beta \) from the row \(A_\beta \). Note that every column of \(B'\) contains at most two nonzero entries. Moreover, if a column has two nonzero entries, then these are 1 and \(-1\). Observe that this implies that \(|\det B'| \in \{0,1\}\). Indeed, if every column of \(B'\) has exactly two nonzero entries, then the rows of \(B'\) sum to the zero vector. Othrewise, expanding \(\det B'\) in a column with at most one nonzero entry shows that \(|\det B'|\) is either zero or equals \(|\det B''|\) for some matrix \(B''\) whose columns have the same property as the columns of \(B'\). \(\square \)
Duality of linear programming and the well-known fact that total unimodularity of the constraint matrix of a linear program guarantees that its optimal value is attained on some integer vector (see [32, Chapter 19]) imply that the maximum of (P) equals the minimum of the following integer version of its dual (D):
where \(h_\sigma = h_\tau = g_\tau := 0\) and, as before, \(z_\tau := -m\).
Suppose that the common value of (P) and (D) is nonzero and let \(g \in \mathbb {Z}_{\ge 0}^P\) and \(h \in \mathbb {Z}^P\) be two feasible vectors achieving the minimum of (D). Choose an arbitrary \(\delta \in P\) with \(g_\delta \ge 1\). It is straightforward to verify that \(g - e_\delta \) and h are a feasible solution to (D) with z replaced by \(z - e_\delta \). Since clearly \(\sum _{\beta \in P} (g-e_\delta )_\beta = \sum _{\beta \in P}g_\beta - 1\), the we may use Claim A.2.1 to conclude that
as claimed.
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Feigin, E. Birational Maps to Grassmannians, Representations and Poset Polytopes. Algebr Represent Theor (2024). https://doi.org/10.1007/s10468-024-10273-x
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DOI: https://doi.org/10.1007/s10468-024-10273-x