Existence and rotatability of the two‐colored Jones–Wenzl projector

The two‐colored Temperley–Lieb algebra 2TLR(sn)$2\,\mathrm{TL}_R({_{s}}{n})$ is a generalization of the Temperley–Lieb algebra. The analogous two‐colored Jones–Wenzl projector JWR(sn)∈2TLR(sn)$\mathrm{JW}_R({_{s}}{n}) \in 2\,\mathrm{TL}_R({_{s}}{n})$ plays an important role in the Elias–Williamson construction of the diagrammatic Hecke category. We give conditions for the existence and rotatability of JWR(sn)$\mathrm{JW}_R({_{s}}{n})$ in terms of the invertibility and vanishing of certain two‐colored quantum binomial coefficients. As a consequence, we prove that Abe's category of Soergel bimodules is equivalent to the diagrammatic Hecke category in complete generality.


HAZI
The algebra 2 TL  (  ) is defined identically, except that the parity conditions on the relations (1) and (2) are swapped.These algebras (introduced by Elias in [3]) form a generalization of the ordinary Temperley-Lieb algebra, which occurs as a special case when [2]  = [2]  .By a standard argument, there is an -basis of 2 TL  (  ) consisting of monomials in the generators   .
The behavior of 2 TL  (  ) is controlled by certain elements []  , []  ∈  for  ∈ ℤ called the two-colored quantum numbers.These elements (defined in (5)) are bivariate polynomials in [2]  and [2]  that are analogous to ordinary quantum numbers.For an integer 0 ⩽  ⩽ , the twocolored quantum binomial coefficient can also be shown to be an element of .Our first main result is the two-colored analogue of the well-known existence theorem for ordinary Jones-Wenzl projectors.
Theorem A. The two-colored Jones-Wenzl projector JW  (  ) exists if and only if is invertible in  for each integer 0 ⩽  ⩽ .
The terminology for two-colored Temperley-Lieb algebras comes from their presentation as diagram algebras.We associate the labels  and  with the colors red and blue, respectively, writing  and  for emphasis.A two-colored Temperley Lieb diagram is a Temperley-Lieb diagram with the planar regions between strands colored with alternating colors.As a diagram, algebra 2 TL  (  ) is spanned by two-colored Temperley-Lieb diagrams with  boundary points on the top and bottom whose leftmost region is colored red.A blue disk inside a red region evaluates to − [2]  , whereas a red disk inside a blue region evaluates to − [2]  .Moreover, we only consider twocolored Temperley-Lieb diagrams up to isotopy.These diagrammatic relations directly correspond to (1)- (4).We draw the two-colored Jones-Wenzl projector as a rectangle labeled JW  (  ): Suppose that both JW  (  ) and JW  (  ) exist.We say that JW  (  ) is rotatable if the (clockwise and counterclockwise) rotations of JW  (  ) by one strand are equal to some scalar multiple of JW  (  ): Our second main result gives a combined condition for the existence and rotatability of two-colored Jones-Wenzl projectors.

Theorem B.
The two-colored Jones-Wenzl projectors JW  (  ) and JW  (  ) exist and are rotatable if and only if The same algebraic condition was first introduced by Abe in the context of the Hecke category [1, Assumption 1.1], which we discuss below.

The Hecke category
The two-colored Temperley-Lieb algebra lies at the heart of the Elias-Williamson diagrammatic Hecke category [5].In more detail, the diagrammatic Hecke category is only well defined when certain two-colored Jones-Wenzl projectors exist and are rotatable.Elias-Williamson initially gave an incorrect algebraic condition for rotatability in [5, (3.3)]; they later identified and partially corrected this error in [6, §5].Our rotatability condition in Theorem B is enough to completely correct this error.

Corollary C. In the absence of parabolic type 𝐻 3 subgroups (see Remark 5.3), the diagrammatic Hecke category is well defined if and only if the underlying realization is an Abe realization (see [1, Assumption 1.1] or Definition 5.1).
Recently, Abe has shown that there is a "bimodule-theoretic" category (a modification of the category of classical Soergel bimodules), which under mild conditions is equivalent to the diagrammatic Hecke category when the underlying realization is an Abe realization [1,2].An important consequence of Corollary C is that this equivalence essentially always holds.

Corollary D.
Assume that Demazure surjectivity holds (see Remark 5.2), and that the base ring is a Noetherian domain.If the diagrammatic Hecke category is well defined, it is equivalent to Abe's category of Soergel bimodules.
We find it noteworthy that our result gives the best possible equivalence result for two seemingly distinct categorifications of the Hecke algebra.

PRELIMINARIES
Let  = ℤ[  ,   ] be the integral polynomial ring in two variables.The two-colored quantum numbers are defined as follows.First, set [1]  = [1]  = 1, [2]  =   , and [2]  =   in .For  > 1, we inductively define These formulas can be rearranged to inductively define []  and []  for  ⩽ 0. For a commutative -algebra , we also define two-colored quantum numbers in  to be the specializations of twocolored quantum numbers in , which we will write in the same way.
These polynomials are bivariate extensions of the usual (one-colored) quantum numbers, which can be recovered as follows.Let  = ∕(  −   ) ≅ ℤ[], where  is the image of   or   .Then, the one-colored quantum number [] is the image of []  or []  in .When  is odd, [] is an even polynomial, so we can formally evaluate [] at  = √     to obtain an element of .When  is even, []∕ [2] is an even polynomial, which we can similarly formally evaluate at  = √     .In both cases, it is easy to show by induction that in .In other words, two-colored quantum numbers are essentially the same as ordinary quantum numbers up to a factor of [2]  and [2]  depending on color.
It is self-evident that the automorphism of  which exchanges   and   ("color swap") also exchanges []  and []  for all .For this reason, we will generally write statements only for []  and leave it to the reader to formulate color-swapped analogues.Similarly, we have 2 TL(  ; [2]  , [2]  ) ≅ 2 TL(  ; [2]  , [2]  ), and this isomorphism maps JW  (  ) to JW  (  ) when they exist, so we will only state our results for 2 TL  (  ) and JW  (  ).
Let  =   1   2 ⋯    be a monomial of length  in the generators of 2 TL(  ).We say that  is reduced if it cannot be rewritten as a monomial   1   2 ⋯    in the generators using (1)-( 4) for some  < .As mentioned in Section 1, the two-colored Temperley-Lieb algebra 2 TL(  ) has a basis consisting of these reduced monomials.As in the one-colored case, there is a bijection between this basis in 2 TL(  ) and (isotopy classes of) two-colored Temperley-Lieb diagrams whose leftmost region is colored red which induces an isomorphism between the algebraic and diagrammatic versions of 2 TL(  ).(For a careful proof of this fact in the one-colored case, see for example, [10,Theorem 2.4].)Under this isomorphism, we have Given an element  ∈ 2 TL  (  ) and a two-colored Temperley-Lieb diagram , we will write coef f ∈  for the coefficient of  when  is written in the diagrammatic basis.
If  is a commutative -algebra for which JW  (  ) exists for all , then the coefficients of JW  (  ) can be calculated inductively as follows.Suppose that  is a two-colored Temperley-Lieb diagram in 2 TL  (  ( + 1)).Let D be the diagram with  + 2 bottom boundary points and  top boundary points obtained by folding down the strand connected to the top right boundary point of .If there is a strand connecting the th and ( + 1)th bottom boundary points of D, let   denote the two-colored Temperley-Lieb diagram with  strands so obtained by deleting this cap.For example, if then and Theorem 2.1.Suppose that JW  (  ) and JW  (  ( + 1)) both exist.Then [ + 1]  is invertible, and we have where the sum is taken over all positions  where   is defined, and  is the color of the deleted cap.
The existence criterion in Theorem A is known to hold in the one-color setting, i.e. when the images of   and   in  are equal.In these circumstances, we write TL  () and JW  () for the one-color Temperley-Lieb algebra and Jones-Wenzl projector.Theorem 2.2 [4,Theorem A.2]. Suppose that  is a commutative -algebra which factors through .Then JW  () exists if and only if the one-color quantum binomial coefficients are invertible in  for all integers 0 ⩽  ⩽ .
In light of the "generic" nature of the coefficients of JW  (  ), we can interpret Theorem 2.2 as description of the denominators of the coefficients of JW Frac  ().Unfortunately, none of the known proofs of this result (most of which use connections to Lie theory in a crucial way) generalize easily to the two-colored setting.
Finally, we will give an alternative criterion for checking rotatability.For  ∈ 2 TL  (  ), define the partial trace of  to be From the definition of the Jones-Wenzl projector, it is easy to see that JW  (  ) is rotatable if and only if pTr(JW  (  )) = 0. Using entirely standard techniques (e.g., [6, §6.6]), one can show that when both JW  (  ) and JW  (  ( − 1)) exist.This gives the following partial rotatability criterion.
The key to proving the full rotatability criterion will be to interpret (7) generically.

PRINCIPAL IDEALS
In this section, we show that several ideals generated by certain two-colored quantum numbers and binomial coefficients are principal.
by [6, (6.5a)-(6.5c)].If  and  are both odd, a similar calculation yields By repeating this step multiple times, we can run Euclid's algorithm, and the result follows.□ Next, we introduce the cyclotomic parts of quantum numbers, which are roughly analogous to cyclotomic polynomials.Recall that the one-color quantum numbers are renormalizations of Chebyshev polynomials of the second kind.More precisely, if we evaluate a quantum number [] at  = 2 cos , we obtain ) .
We define the cyclotomic part of the one-color quantum number [] to be the polynomial ) .
Proof.Both (i) and (ii) follow from the definition and basic properties of cyclotomic fields and algebraic integers.Applying Möbius inversion to (ii) yields (iii).For the final claim, we observe that if  > 2 then Θ  is an even polynomial, so is of the form of Ψ  ( 2 ) for some and ℚ(2 cos(2∕) + 2) = ℚ(cos(2∕)) is a field extension of ℚ of degree ()∕2, Ψ  must be the minimal polynomial of 4 cos 2 (∕).□ Definition 3.3.For  ∈ ℕ, we define the cyclotomic part of the two-colored quantum number []  to be Using (6)  For  ∈  and  > 1 an integer, we define the cyclotomic valuation  , () to be the exponent of the highest power of Θ , dividing .This extends to Frac  in the obvious way; namely, we define  , (∕g) =  , () −  , (g) for , g ∈ .If  and g are products of -colored cyclotomic parts, then (∕g) .
For  ∈ , we similarly define   () to be the highest power of Θ  dividing , which extends in a completely analogous way to Frac .
Lemma 3.7.Let ,  be nonnegative integers.For all integers 1 <  ⩽ , we have In particular,  , Proof.We will prove the result by induction.Let and write Suppose that we have shown that   is principal, generated by []  > .We will show that  +1 is principal, generated by []  >+1 .It is enough to show that Clearly and we must have  +  + 1 ⩾ .Now let  = gcd(,  − ( + 1)).As  + 1| and  + 1|, we have  + 1|, and in particular  + 1 ⩽ .We also have |, so in particular  −  ⩾ .We combine these two equalities to obtain  +  + 1 ⩽ , with equality if and only if  −  =  and  + 1 = .This immediately implies that |, so Θ , does not divide . □ Theorem 3.9.Let  ∈ ℕ.The fractional ideal of  generated by Proof.We follow a similar strategy as in the proof of Theorem 3.8.Let   denote the fractional ideal generated by and let |−+ for some 1⩽⩽ ∤+1 Θ , .

EXISTENCE AND ROTATABILITY
Let  = Frac  and  = Frac .Our goal in this section is to prove Theorem A by showing that the denominators of the coefficients of JW  (  ) divide Θ , by comparing them with the coefficients of JW  ().First, we prove the analogous statement for JW  ().Proof.We proceed by induction.Suppose that the result holds for  = , and let  be a onecolored Temperley-Lieb diagram in TL  ( + 1).By the one-color version of Theorem 2.1, we have If  ∤  + 1, then   ([]∕[ + 1]) ⩾ 0 for any  and   (coef f ∈JW  ()   ) ⩾ −1.On the other hand, In either case, the sum of the two valuations is at least −1, so the right-hand side of ( 10) is at least −1.Now suppose that we have equality.By Theorem 2.2, the one-color Jones-Wenzl projector exists over the subring .
The natural embedding  binom ⊂  induces an embedding TL  binom ( + 1) → TL  ( + 1), and the image of JW  binom ( + 1) in TL  ( + 1) is clearly a Jones-Wenzl projector.Since Jones-Wenzl projectors are unique, we conclude that the coefficients of JW  ( + 1) lie in  binom .In particular, if the -valuation of any given coefficient is negative, then Θ  must divide for some 0 ⩽  ⩽  + 1, so it must divide the least common multiple of . But a consequence of the one-color version of Theorem 3.9 is that this least common multiple is So, we must have (Θ  , g +1 ) = (Θ  ), so 1 <  ⩽  + 1 and  ∤  + 2 as required.□ Now let  ′ = []∕( 2 −     ).We view  ′ as both an -algebra and an -algebra in the obvious way.Writing  ′ = Frac  ′ , we have an isomorphism In all cases, the sum of the two valuations is at least −1 (and at least 0 in the case where  = ), so the right-hand side of (13) is at least −1.

]
−1 for some  ∈  and some integer 0 ⩽  ⩽  − 1.Now observe that and similarly for , we conclude that  vanish for all integers 1 ⩽  ⩽ .□

APPLICATIONS TO THE HECKE CATEGORY
The diagrammatic Hecke category  of Elias-Williamson is constructed from a reflection representation of a Coxeter group called a realization.For each finite parabolic dihedral subgroup they identify a corresponding two-colored Temperley-Lieb algebra, whose defining parameters depend on the realization [5, §5.2].In [6, §5], Elias-Williamson highlight some hidden assumptions about their realizations from [5].Their most basic assumption (without which the diagrammatic Hecke category is not well defined) is that certain two-colored Jones-Wenzl projectors exist and are rotatable.For the benefit of future work, we give a corrected definition of a realization (which we call an Abe realization) that ensures the existence and rotatability of these Jones-Wenzl projectors. is enough to ensure the existence and rotatability of JW  (  (  − 1)).(This error was identified in [6] but only partially resolved there.)In the same paper, Elias-Williamson also incorrectly state that ( 16) is equivalent to (15).Amusingly, when these two statements are combined these errors accidentally cancel and the resulting statement is equivalent to Corollary C!