# Crystals¶

Let $$T$$ be a CartanType with index set $$I$$, and $$W$$ be a realization of the type $$T$$ weight lattice.

A type $$T$$ crystal $$C$$ is a colored oriented graph equipped with a weight function from the nodes to some realization of the type $$T$$ weight lattice such that:

• Each edge is colored with a label in $$i \in I$$.

• For each $$i\in I$$, each node $$x$$ has:

• at most one $$i$$-successor $$f_i(x)$$;
• at most one $$i$$-predecessor $$e_i(x)$$.

Furthermore, when they exist,

• $$f_i(x)$$.weight() = x.weight() - $$\alpha_i$$;
• $$e_i(x)$$.weight() = x.weight() + $$\alpha_i$$.

This crystal actually models a representation of a Lie algebra if it satisfies some further local conditions due to Stembridge [St2003].

REFERENCES:

 [St2003] J. Stembridge, A local characterization of simply-laced crystals, Trans. Amer. Math. Soc. 355 (2003), no. 12, 4807-4823.

EXAMPLES:

We construct the type $$A_5$$ crystal on letters (or in representation theoretic terms, the highest weight crystal of type $$A_5$$ corresponding to the highest weight $$\Lambda_1$$):

sage: C = CrystalOfLetters(['A',5]); C
The crystal of letters for type ['A', 5]


It has a single highest weight element:

sage: C.highest_weight_vectors()
[1]


A crystal is an enumerated set (see EnumeratedSets); and we can count and list its elements in the usual way:

sage: C.cardinality()
6
sage: C.list()
[1, 2, 3, 4, 5, 6]


as well as use it in for loops:

sage: [x for x in C]
[1, 2, 3, 4, 5, 6]


Here are some more elaborate crystals (see their respective documentations):

sage: Tens = TensorProductOfCrystals(C, C)
sage: Spin = CrystalOfSpins(['B', 3])
sage: Tab  = CrystalOfTableaux(['A', 3], shape = [2,1,1])
sage: Fast = FastCrystal(['B', 2], shape = [3/2, 1/2])
sage: KR = KirillovReshetikhinCrystal(['A',2,1],1,1)


One can get (currently) crude plotting via:

sage: Tab.plot()


If dot2tex is installed, one can obtain nice latex pictures via:

sage: K = KirillovReshetikhinCrystal(['A',3,1], 1,1)
sage: view(K, pdflatex=True, tightpage=True) #optional - dot2tex graphviz


or with colored edges:

sage: K = KirillovReshetikhinCrystal(['A',3,1], 1,1)
sage: G = K.digraph()
sage: G.set_latex_options(color_by_label = {0:"black", 1:"red", 2:"blue", 3:"green"}) #optional - dot2tex graphviz
sage: view(G, pdflatex=True, tightpage=True) #optional - dot2tex graphviz


For rank two crystals, there is an alternative method of getting metapost pictures. For more information see C.metapost?

Todo

• Vocabulary and conventions:
• For a classical crystal: connected / highest weight / irreducible
• ...
• Layout instructions for plot() for rank 2 types
• RestrictionOfCrystal

Most of the above features (except Littelmann/alcove paths) are in MuPAD-Combinat (see lib/COMBINAT/crystals.mu), which could provide inspiration.

class sage.combinat.crystals.crystals.CrystalBacktracker(crystal, index_set=None)

Time complexity: $$O(nF)$$ amortized for each produced element, where $$n$$ is the size of the index set, and $$F$$ is the cost of computing $$e$$ and $$f$$ operators.

Memory complexity: $$O(D)$$ where $$D$$ is the depth of the crystal.

Principle of the algorithm:

Let $$C$$ be a classical crystal. It’s an acyclic graph where all connected component has a unique element without predecessors (the highest weight element for this component). Let’s assume for simplicity that $$C$$ is irreducible (i.e. connected) with highest weight element $$u$$.

One can define a natural spanning tree of $$C$$ by taking $$u$$ as the root of the tree, and for any other element $$y$$ taking as ancestor the element $$x$$ such that there is an $$i$$-arrow from $$x$$ to $$y$$ with $$i$$ minimal. Then, a path from $$u$$ to $$y$$ describes the lexicographically smallest sequence $$i_1,\dots,i_k$$ such that $$(f_{i_k} \circ f_{i_1})(u)=y$$.

Morally, the iterator implemented below just does a depth first search walk through this spanning tree. In practice, this can be achieved recursively as follow: take an element $$x$$, and consider in turn each successor $$y = f_i(x)$$, ignoring those such that $$y = f_j(x^{\prime})$$ for some $$x^{\prime}$$ and $$j<i$$ (this can be tested by computing $$e_j(y)$$ for $$j<i$$).

EXAMPLES:

sage: from sage.combinat.crystals.crystals import CrystalBacktracker
sage: C = CrystalOfTableaux(['B',3],shape=[3,2,1])
sage: CB = CrystalBacktracker(C)
sage: len(list(CB))
1617
sage: CB = CrystalBacktracker(C, [1,2])
sage: len(list(CB))
8


Affine Crystals

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Direct Sum of Crystals