OCAMI Algebra Seminar Demonstration¶

In [4]:

load("tors_lattice.py")

Input your poset of torsion classes!¶

Construct your poset and name your poset poset.

Input using SageMath's Poset function.

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poset = Poset( ([1,2,3,4,5], [[1,2], [2,5], [1,3],[3,4],[4,5]] ) )

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poset

Out[6]:

Important remark¶

The directions of Hasse arrows in SageMath is opposite to our convention!

Input using Jan Geuenich's String Applet and my String Applet to Sage converter

In [7]:

load("converter.py")

In [9]:

poset = Poset( SAtoSage("data.tex"))

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myshow(poset, label = False)

Enjoy!¶

Construct the lattice of torsion classes by TorsLattice(poset)

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tors = TorsLattice(poset)

The lattice of wide subcats: tors.wide_lattice()

In [12]:

wide = tors.wide_lattice()

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wide

Out[13]:

Finite lattice containing 60 elements (use the .plot() method to plot)

In [14]:

myshow(wide)

The lattice of ICE-closed subcats: tors.ice_lattice()

In [16]:

myshow(tors.ice_lattice(), label = False)

The poset of torsion hearts: tors.heart_poset()

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heart = tors.heart_poset()

In [18]:

heart.is_lattice()

Out[18]:

False

$\Delta(\Lambda)$: tors.s_tau_tilt_complex()

In [19]:

cpx = tors.s_tau_tilt_complex()

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cpx.h_vector()

Out[21]:

[1, 14, 30, 14, 1]

Path algebra¶

$\mathsf{tors} kQ$ for a Dynkin quiver $Q$ is given by the Cambiran lattice [Ingalls-Thomas].

Consider the path algebra $k [ 1 \to 2 \leftarrow 3]$, type $A_3$ with its source sequence $(1,3,2)$.

Name W = WeylGroup(["A3"]), and $\mathsf{tors} kQ$ is W.cambrian_lattice((1,3,2)).

In [22]:

W = WeylGroup(["A3"])

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poset = W.cambrian_lattice((1,3,2))

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tors = TorsLattice(poset)

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myshow(tors, label = False)

$\mathsf{wide} kQ$ should be isomorphic to the so-called non-crossing partition lattice, given by posets.NoncrossingPartitions(W).

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wide = tors.wide_lattice()

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NC = posets.NoncrossingPartitions(W)

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wide.is_isomorphic(NC)

Out[29]:

True

$\Delta(kQ)$ should be isomorphic to ClusterComplex(["A3"]).

In [30]:

cpx = tors.s_tau_tilt_complex()

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cpx.is_isomorphic(ClusterComplex(["A3"]))

Out[31]:

True

Preprojective algebra¶

$\mathsf{tors} \Pi$ is given by the right weak order of the Weyl group [Mizuno].

Consider the preprojective algebra of $\Pi$ type $D_4$.

Let's google how to input this!

In [32]:

W = CoxeterGroup(["D4"])

In [33]:

poset = W.weak_poset()
tors = TorsLattice(poset)

In [34]:

myshow(tors, label = False)