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All tight monomials in quantum group for type
A
_{5} with
t ≤ 6 are determined in this paper.

The term “quantum groups” was popularized by V. G. Drinfel’d in his address to the International Congress of Mathematicians (ICM) in Berkeley (1986). However, quantum groups are actually not groups; they are nontrivial deformations of the universal enveloping algebras of semisimple Lie algebras, also called quantized enveloping algebras. These algebras were introduced independently by Drinfel’d [

The positive part of a quantum group has a kind of important basis, i.e., canonical basis introduced by Lusztig [_{n}. Lusztig firstly introduced algebraic definition of canonical basis of quantum groups for the simply laced case (i.e., A_{n}, D_{n}, E_{n}), and gave explicitly the longest monomials for type A_{1}, A_{2}, which were all of canonical basis elements (see [_{3} and computed 8 longest tight monomials of type A_{3}. He also asked when we had (semi)tightness in type A_{n}. Based on Lusztig’s work, Xi [_{3} (consisting of 8 longest monomials and 6 polynomials with one-dimensional support). For type A_{4}, Hu, Ye and Yue [_{n} (n ≥ 5), Marsh [_{5}. However, he proved that a class of special longest monomials did not satisfy sufficient condition of tightness or semitightness for type A_{n} (n ≥ 6) (although it might turn out that the corresponding monomials were still tight). Reineke [_{3}, in which 8 longest tight monomials were the same as Lusztig and Xi’s results.

Based on Reineke’s criterion and some other results, all tight monomials for type A_{5} with t ≤ 6 are determined in this paper.

Let

tries making the matrix DC symmetric. Let ^{+} is the

where^{+} be the

-subalgebra of U^{+} generated by_{i} of U^{+}. Lusztig proved that the _{i} is independent of the choice of i, write_{i} in the ^{+} de- fined by

A monomial in U^{+} is an element of the form

where

Let _{0} and arrow set Q_{1}. Write _{ρ}

and t_{ρ} denote the head and the tail of ρ respectively. An automorphism σ of Q is a permutation on the vertices of

Q and on the arrows of Q such that

automorphism σ as_{0} and Q_{1}, respectively. The valuation of

is given by

Euler form of

where

Let t be a non-negative integer. Let

Define

where

Obviously,

The following results are very useful in the determination of tight monomials.

Theorem 2.1 [

Theorem 2.2 [_{n}, D_{n}, E_{n},

If

Corollary 2.3. When

Corollary 2.4. If

and any mutually different

is also tight.

Theorem 2.5 [

(a) For

(b) For

Theorem 2.6 [^{+} induced by Dynkin diagram

automorphism of

If

Let

as a word

a monomial

By Theorem 2.5(b), we only consider those words _{1} (or i_{t}) in the front (or behind) of i_{1} (or i_{t}), secondly delete the words with t-value, lastly apply the automorphism

Theorem 3.1. Let M_{t} be the set of all tight monomials with t-value in quantum group for type A_{5}, we have the following results.

(1) t = 0,

(2) t = 1, if

(3) t = 2, if

(4) t = 3, if

then

(5) t = 4, if

then

(6) t = 5, if

then

(7) If t = 6,

then

Consider the quiver _{5}, where

Symmetric Euler form

where

By simple computation, we have

Let us prove Theorem 3.1.

Case 1.

Case 2. t = 3. Applying the word-procedure on S_{2}, we get 33 words with 3-value. By considering

and

Obviously,

Case 3. t = 4. Applying the word-procedure on

where

and

Obviously,

are all not tight for any

Monomials in

where

and

Case 4. t = 5. Applying the word-procedure on

For

where

and

is tight by Theorem 2.2.

For

where

and

is tight by Theorem 2.2.

For

where

and

is tight by Theorem 2.2.

For

where

and

is tight by Theorem 2.2.

Case 5. t = 6. Applying the word-procedure on S_{5}, and deleting words including subwords 1212, 2121, 2323, 3232, 3434, 4343, 4545 and 5454(considering Theorem 2.5(a)), we get 228 words with 6-value. By considering Φ and Ψ, we get

where

and

are all not tight for any

By Corollary 2.4, we have

For

where

and

is tight by Theorem 2.2.

For

where

and

is tight by Theorem 2.2.

For

where

and

is tight by Theorem 2.2.

For

where

and

is tight by Theorem 2.2.

This paper is supported by the NSF of China (No. 11471333) and Basic and advanced technology research project of Henan Province (142300410449).

YuwangHu,GuiweiLi,JunWang, (2015) Tight Monomials in Quantum Group for Type A_{5} witht ≤ 6. Advances in Linear Algebra & Matrix Theory,05,63-75. doi: 10.4236/alamt.2015.53007