Multiplicative closure of the standard monomials for the Temperley–Lieb algebras of types B and D

For the Temperley–Lieb algebras $T(B_n)$ and $T(D_n)$, Kim and Lee constructed Gröbner–Shirshov bases and the corresponding standard monomials, and asserted — supporting it with worked examples only — that the product of two standard monomials is again a scalar multiple of a single standard monomial. This closure is an immediate consequence of the reduction, because every relation of the fixed presentations and every element of the Gröbner–Shirshov bases is a binomial, a word equal to a scalar multiple of a word; it is therefore not itself the research problem. The problem is instead to determine in general — both types, every rank $n$, every pair of standard monomials, with respect to the degree-lexicographic order and Gröbner–Shirshov bases fixed in that work — the resulting standard monomial $w(u,v)$ and scalar $\alpha(u,v,\delta)$ with $uv=\alpha\, w$ in closed form; no source determines these multiplicative structure constants of the basis.

Background

The Temperley–Lieb algebra first appeared in 1971 in Temperley and Lieb's exact solution of a lattice percolation model, and later reappeared in knot theory as a quotient of the type A Hecke algebra (the finite-dimensional quotient of the braid-group algebra governing the one-variable link polynomials). In its classical type $A_{n-1}$ form it is the complex algebra on generators $E_1,\dots,E_{n-1}$ with relations $E_i^2=\delta E_i$, $E_iE_j=E_jE_i$ for $|i-j|>1$, and $E_iE_jE_i=E_i$ for $|i-j|=1$, where $\delta\in\mathbb{C}$ is a parameter. Analogues for other finite Coxeter types follow the same pattern: one generator per node of the Coxeter diagram (the graph encoding the simple reflections of a finite reflection group), with quadratic relations, commutation for non-adjacent nodes, and braid relations truncated to length three or four for adjacent nodes. Explicitly, for $n\ge 2$ the algebra $T(B_n)$ has generators $E_0,\dots,E_{n-1}$ and relations $E_i^2=\delta E_i$ ($0\le i\le n-1$), $E_iE_j=E_jE_i$ for $i>j+1$, $E_iE_jE_i=E_i$ for $j=i\pm 1$ with $i,j>0$, and $E_iE_jE_iE_j=2E_iE_j$ for $\{i,j\}=\{0,1\}$; for $n\ge 4$ the algebra $T(D_n)$ has the same quadratic and far-commutation relations ($E_iE_0=E_0E_i$ for $i\ne 2$, $E_iE_j=E_jE_i$ for $1<j+1<i\le n-1$) together with $E_iE_jE_i=E_i$ for $j=i\pm 1$ with $i,j>0$ and $E_iE_jE_i=E_i$ for $\{i,j\}=\{0,2\}$. Throughout, $E_{i,j}$ denotes the descending word $E_iE_{i-1}\cdots E_j$ for $i\ge j$ (in type $D$, $E_{i,0}=E_iE_{i-1}\cdots E_3E_2E_0$ for $i\ge 2$ and $E_{1,0}=E_0$), $E^{i,j}$ denotes the ascending word $E_iE_{i+1}\cdots E_j$ for $i\le j$, the empty-word conventions $E_{i,i+1}=E^{i+1,i}=1$ are used, and one writes $E'_{i,j}=E_{i,0}E^{1,j}$.

The tool for studying these algebras here is Gröbner–Shirshov basis theory, the noncommutative analogue of Gröbner bases for polynomial rings. Given a set $S$ of monic polynomials in the free associative algebra $F\langle X\rangle$ and a monomial order (a well-order on words compatible with multiplication), a word is $S$-standard (or reduced) if it contains no leading monomial of any element of $S$ as a subword; Shirshov's Composition Lemma, in equivalent form Bergman's diamond lemma, states that if all compositions (overlaps and inclusions of leading monomials) reduce to zero, then $S$ is a Gröbner–Shirshov basis and the $S$-standard monomials form a linear basis of the quotient algebra, so every element has a unique normal form as a linear combination of standard monomials. For type $A_{n-1}$ this was carried out by Kang, Lee, Lee and Oh: the standard monomials are the products $E_{i_1,j_1}\cdots E_{i_p,j_p}$ with $1\le i_1<\cdots<i_p\le n-1$, $1\le j_1<\cdots<j_p\le n-1$ and $i_k\ge j_k$, and their number is the Catalan number $C_n=\frac{1}{n+1}\binom{2n}{n}$, which is also the dimension of the algebra. The dimensions in types B and D were computed by Fan and by Stembridge through the enumeration of fully commutative elements — elements of the Coxeter group none of whose reduced words contains an alternating subword $sts$ with $s,t$ non-commuting generators, equivalently elements whose reduced expressions are all connected by commuting relations alone: $\dim T(B_n)=(n+2)C_n-1$ and $\dim T(D_n)=\frac{n+3}{2}C_n-1$.

Kim and Lee extended the type A Gröbner–Shirshov computation to types B and D. With the degree-lexicographic order $E_0<E_1<\cdots<E_{n-1}$ (words compared first by length, then lexicographically), completing the defining relations under composition yields finite Gröbner–Shirshov bases $\hat{R}_T(B_n)$ and $\hat{R}_T(D_n)$ (Theorems 4.2 and 5.2 of their paper); the standard monomials are the type A ones together with explicitly parametrized families containing the generator $E_0$ — in type B, monomials of the forms $E_0E_{i_1,j_1}\cdots E_{i_p,j_p}$ and $E'_{i_1,j_1}E_{i_2,j_2}\cdots E_{i_p,j_p}$ with nested index conditions, and analogously in type D. Counting these monomials reproduces the Fan–Stembridge dimensions, which is how the authors certify the Gröbner–Shirshov property. Among the properties of the construction they highlight: the standard monomials lie inside the Temperley–Lieb algebra itself (unlike the fully commutative elements, which live in the underlying Coxeter group), and, third, that the product of two standard monomials becomes a standard monomial up to a scalar multiple — a property they support only with worked examples, such as $E_2(E_0E_{1,0}E_{2,0})=\delta E_0E_{2,0}$ in $T(B_3)$ and $E_3(E_0E_{2,1}E_{3,0})=E'_1E_3$ in $T(D_4)$. It holds in general, however, as an immediate consequence of the reduction, since the defining relations and the completed Gröbner–Shirshov bases consist of binomials. Such a property would make the standard monomials a multiplicative basis — a basis whose pairwise products are scalar multiples of basis elements — which is the role played by the planar-diagram basis in the classical type A theory and by the diagram basis of the blob algebra (a diagrammatic relative of the type B Temperley–Lieb algebra built from planar matchings decorated with marked arcs) in type B.

Problem Statement

Fix the setting of Kim and Lee (arXiv:1808.05026): the Temperley–Lieb algebra $T(B_n)$ for $n\ge 2$ and $T(D_n)$ for $n\ge 4$ with the presentations given in the Background; the degree-lexicographic monomial order $E_0<E_1<\cdots<E_{n-1}$; the Gröbner–Shirshov bases $\hat{R}_T(B_n)$ and $\hat{R}_T(D_n)$ of their Theorems 4.2 and 5.2; and the corresponding sets $M_T(B_n)$ and $M_T(D_n)$ of standard monomials (the words containing no leading monomial of the basis as a subword, explicitly parametrized in their Sections 4 and 5).

1. Determine in closed form, for both types and every rank $n$ in the stated range, the standard monomial $w(u,v)$ and scalar $\alpha(u,v,\delta)$ with $uv=\alpha\, w$ for two standard monomials $u,v$ of the same type — the multiplicative structure constants of the standard-monomial basis. The scalars arising in such products are built from powers of $\delta$ and factors of $2$ coming from the $E_0$-relations, the surviving standard monomial and scalar are read off from the chain-independent normal form of $uv$ obtained by the reduction, and no source determines $w(u,v)$ and $\alpha(u,v,\delta)$ in closed form.

References

1. Sungsoon Kim and Dong-il Lee, Gröbner–Shirshov bases for Temperley–Lieb algebras of types B and D, Journal of Algebra and Its Applications 19 (2020), no. 1, 2050002, DOI: 10.1142/S0219498820500024 (arXiv:1808.05026; zbMATH Zbl 1481.20013), https://arxiv.org/abs/1808.05026
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